Pre-Calc Trigonometry
|
|
|
- Esmond Carr
- 5 years ago
- Views:
Transcription
1 Slide 1 / 207
2 Slide 2 / 207 Pre-Calc Trigonometry
3 Slide 3 / 207 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Power Reducing Sum to Product Product to Sum Inverse Trig Functions Trig Equations click on the topic to go to that section
4 Slide 4 / 207 Unit Circle Return to Table of Contents
5 Slide 5 / 207 Unit Circle Goals and Objectives Students will understand how to use the Unit Circle to find angles and determine their trigonometric value.
6 Slide 6 / 207 Unit Circle Why do we need this? The Unit Circle is a tool that allows us to determine the location of any angle.
7 Slide 7 / 207 Unit Circle Special Right Triangles
8 Slide 8 / 207 Unit Circle Example 1: Find a Example 2: Find b & c Teacher 6 a 4 c b
9 Slide 9 / 207 Unit Circle Example 3: Find d Example 4: Find e Teacher d 8 9 e
10 Slide 10 / 207 Unit Circle Example 5: Find f Example 6: Find g & h Teacher f 1 g h 1
11 Slide 11 / 207 Unit Circle 30 o 45 o 60 o 60 o 45 o 30 o 30 o 30 o 45 o 60 o 60 o 45 o
12 Unit Circle Slide 12 / 207
13 Unit Circle Slide 13 / 207
14 Unit Circle Slide 14 / 207
15 Unit Circle Slide 15 / 207
16 Slide 16 / 207
17 Slide 17 / 207
18 Slide 18 / 207
19 Slide 19 / 207
20 Slide 20 / 207
21 Slide 21 / 207
22 Slide 22 / 207 Unit Circle 4 Which function is positive in the second quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
23 Slide 23 / 207 Unit Circle 5 Which function is positive in the fourth quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
24 Slide 24 / 207 Unit Circle 6 Which function is positive in the third quadrant? Choose all that apply. Teacher A B C D E F cos x sin x tan x sec x csc x cot x
25 Slide 25 / 207 Unit Circle Example: Given the terminal point of ( -5 / 13, -12 / 13 ) find sin x, cos x, and tan x. Teacher
26 Slide 26 / 207 Unit Circle 7 Given the terminal point find tan x. Teacher
27 Slide 27 / 207 Unit Circle 8 Given the terminal point find sin x. Teacher
28 Slide 28 / 207 Unit Circle 9 Given the terminal point find tan x. Teacher
29 Slide 29 / 207 Unit Circle 10 Knowing sin x = Teacher Find cos x if the terminal point is in the first quadrant
30 Slide 30 / 207 Unit Circle 11 Knowing sin x = Teacher Find cos x if the terminal point is in the 2 nd quadrant
31 Slide 31 / 207
32 Slide 32 / 207 Graphing Return to Table of Contents
33 Slide 33 / 207 Graphing Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
34 Slide 34 / 207 Graphing Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
35 Slide 35 / 207 Graphing Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
36 Graphing Slide 36 / 207 Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
37 Slide 37 / 207 Graphing Parts of a trig graph cos x Amplitude x Period
38 Slide 38 / 207
39 Slide 39 / 207 Graphing y= a sin(x) or y= a cos(x) In the study of transforming parent functions, we learned "a" was a vertical stretch or shrink. For trig functions it is called the amplitude.
40 Slide 40 / 207 Graphing In y= cos(x), a=1 This means at any time, y= cos (x) is at most 1 away from the axis it is oscillating about. Teacher Find the amplitude: y= 3 sin(x) y= 2 cos(x) y= -4 sin(x)
41 Slide 41 / 207 Graphing 13 What is the amplitude of y = 3cosx? Teacher
42 Slide 42 / 207 Graphing 14 What is the amplitude of y = 0.25cosx? Teacher
43 Slide 43 / 207 Graphing 15 What is the amplitude of y = -sinx? Teacher
44 Slide 44 / 207
45 Slide 45 / 207 Graphing y= sin b(x) or y= cos b(x) In the study of transforming parent functions, we learned "b" was a horizontal stretch or shrink. y= cos x has b=1. Therefore cos x can make one complete cycle is 2#. For trig functions it is called the period.
46 Slide 46 / 207 Graphing y = cos x completes 1 "cycle" in 2#. So the period is 2π. y = cos 2x completes 2 "cycles" in 2# or 1 "cycle" in #. The period is # y = cos 0.5x completes 1 / 2 a cycle in 2#. The period is 4#.
47 Slide 47 / 207 Graphing The period for y= cos bx or y= sin bx is
48 Slide 48 / 207 Graphing 16 What is the period of Teacher A B C D
49 Slide 49 / 207 Graphing 17 What is the period of Teacher A B C D
50 Slide 50 / 207 Graphing 18 What is the period of Teacher A B C D
51 Slide 51 / 207
52 Slide 52 / 207 Graphing y= sin (x+c) or y= cos (x+c) In the study of transforming parent functions, we learned "c" was a horizontal shift y= cos (x+# ) has c = π. The graph of y= cos (x+π) is the graph of y=cos(x) shifted to the left #. For trig functions it is called the phase shift.
53 Slide 53 / 207
54 Slide 54 / 207
55 Slide 55 / 207
56 Slide 56 / 207
57 Slide 57 / 207
58 Slide 58 / 207
59 Slide 59 / 207 Graphing y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift
60 Slide 60 / 207 Graphing 23 What is the vertical shift in Teacher
61 Slide 61 / 207 Graphing 24 What is the vertical shift in Teacher
62 Slide 62 / 207 Graphing 25 What is the vertical shift in Teacher
63 Slide 63 / 207
64 Slide 64 / 207
65 Slide 65 / 207
66 Slide 66 / 207
67 Slide 67 / 207
68 Slide 68 / 207
69 Slide 69 / 207 Graphing 30 What is the amplitude of this cosine graph? Teacher
70 Slide 70 / 207 Graphing 31 What is the period of this cosine graph? (use 3.14 for pi) Teacher
71 Slide 71 / 207 Graphing 32 What is the phase shift of this cosine graph? Teacher
72 Slide 72 / 207 Graphing 33 What is the vertical shift of this cosine graph? Teacher
73 Slide 73 / 207 Graphing 34 Which of the following of the following are equations for the graph? Teacher A B C D
74 Slide 74 / 207 Law of Sines Return to Table of Contents
75 Slide 75 / 207
76 Slide 76 / 207 Law of Sines When to use Law of Sines (Recall triangle congruence statements) ASA AAS SAS (use Law of Cosines) SSS (use Law of Cosines) SSA (use Law of Sines- but be cautious!)
77 Slide 77 / 207
78 Slide 78 / 207
79 Slide 79 / 207 Law of Sines Example: Teddy is driving toward the Old Man of the Mountain, the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain? Teacher y x
80 Slide 80 / 207
81 Slide 81 / 207
82 Slide 82 / 207 Law of Sines with SSA. SSA information will lead to 0, 1,or 2 possible solutions. The one solution answer comes from when the bigger given side is opposite the given angle. The 2 solution and no solution come from when sin -1 is used in the problem and the answer and its supplement are evaluated, sometimes both will work, sometimes one will work,and sometimes neither will work.
83 Slide 83 / 207 Law of Sines Example B solve triangle ABC Teacher A C
84 Slide 84 / 207 Law of Sines Example B solve triangle ABC Teacher A C
85 Slide 85 / 207 Law of Sines Solution 1 B Solution 2 B Teacher A C A C 5
86 Slide 86 / 207 Law of Sines Example B solve triangle ABC Teacher 14 7 A 50 C
87 Slide 87 / 207
88 Slide 88 / 207 Law of Sines 38 How many triangles meet the following conditions? Teacher
89 Slide 89 / 207 Law of Sines 39 How many triangles meet the following conditions? Teacher
90 Slide 90 / 207 Law of Cosines Return to Table of Contents
91 Slide 91 / 207
92 Slide 92 / 207 Law of Cosines When we began to study Law of Sines, we looked at this table: When to use Law of Sines (Recall triangle congruence statements) ASA AAS SAS (use Law of Cosines) SSS (use Law of Cosines) SSA (use Law of Sines- but be cautious!) Its now time to look at SAS and SSS triangles.
93 Slide 93 / 207
94 Slide 94 / 207
95 Slide 95 / 207
96 Slide 96 / 207
97 Slide 97 / 207 Law of Cosines Example: Joe went camping. Sitting at his camp site he noticed it was 3 miles to one end of the lake and 4 miles to the other end. He determined that the angle between these two line of sites is 105 degrees. How far is it across the lake? Teacher x
98 Slide 98 / 207
99 Slide 99 / 207
100 Slide 100 / 207
101 Slide 101 / 207 Identities Return to Table of Contents
102 Slide 102 / 207 Trigonometry Identities are useful for simplifying expressions and proving other identities.
103 Slide 103 / 207 Pythagorean Identities Return to Table of Contents
104 Slide 104 / 207 Pythagorean Identities Trigonometric Ratios
105 Slide 105 / 207 Pythagorean Identities Pythagorean Identities
106 Slide 106 / 207 Pythagorean Identities Simplify: Teacher
107 Slide 107 / 207 Pythagorean Identities Simplify: Teacher
108 Slide 108 / 207 Pythagorean Identities Simplify: Teacher
109 Slide 109 / 207 Pythagorean Identities Prove: Teacher
110 Slide 110 / 207
111 Slide 111 / 207 Pythagorean Identities Prove: Teacher
112 Slide 112 / 207
113 Slide 113 / 207 Pythagorean Identities 43 The following expression can be simplified to which choice? Teacher A B C D
114 Slide 114 / 207 Pythagorean Identities 44 The following expression can be simplified to which choice? Teacher A B C D
115 Slide 115 / 207 Pythagorean Identities 45 The following expression can be simplified to which choice? Teacher A B C D
116 Slide 116 / 207 Angle Sum/Difference Return to Table of Contents
117 Slide 117 / 207 Angle Sum/Difference Angle Sum/Difference Identities are used to convert angles we aren't familiar with to ones we are (ie. multiples of 30, 45, 60, & 90).
118 Slide 118 / 207 Angle Sum/Difference Sum/ Difference Identities
119 Slide 119 / 207 Angle Sum/Difference Find the exact value of Teacher
120 Slide 120 / 207
121 Slide 121 / 207 Angle Sum/Difference Find the exact value of Teacher
122 Slide 122 / 207 Angle Sum/Difference Find the exact value of Teacher
123 Slide 123 / 207 Angle Sum/Difference Prove: Teacher
124 Slide 124 / 207 Angle Sum/Difference Prove: Teacher
125 Slide 125 / 207 Angle Sum/Difference 46 Which choice is another way to write the given expression? Teacher A B C D
126 Slide 126 / 207 Angle Sum/Difference 47 Which choice is the exact value of the given expression? Teacher A B C D
127 Slide 127 / 207 Double Angle Return to Table of Contents
128 Slide 128 / 207 Double Angle Double-Angle Identities
129 Slide 129 / 207
130 Slide 130 / 207 Double Angle Write cos3x in terms of cosx Teacher
131 Slide 131 / 207 Double Angle 48 Which of the following choices is equivalent to the given expression? Teacher A B C D
132 Slide 132 / 207
133 Slide 133 / 207 Double Angle 50 Which of the following choices is equivalent to the given expression? Teacher A B C D
134 Slide 134 / 207 Half Angle Return to Table of Contents
135 Slide 135 / 207
136 Slide 136 / 207 Half Angle Find the exact value of cos15 using Half-Angle Identity Teacher
137 Slide 137 / 207 Half Angle Find the exact value of tan 22.5 Teacher
138 Slide 138 / 207 Half Angle 51 Find the exact value of Teacher A B C D
139 Slide 139 / 207 Half Angle 52 Find the exact value of Teacher A B C D
140 Slide 140 / 207 Half Angle Find cos( u / 2 ) if sin u= - 3 / 7 and u is in the third quadrant Pythagorean Identity but Why Negative? Teacher
141 Slide 141 / 207
142 Slide 142 / 207 Half Angle 54 Find if and u is in the 4th quadrant? A Teacher B C D
143 Slide 143 / 207 Power Reducing Identities Return to Table of Contents
144 Slide 144 / 207 Power Reducing Identities Power Reducing Identities
145 Slide 145 / 207 Power Reducing Identities Reduce sin 4 x to an expression in terms of first power cosines. Teacher
146 Slide 146 / 207 Power Reducing Identities Reduce cos 4 x to an expression in terms of first power cosines. Teacher
147 Slide 147 / 207 Power Reducing Identities 55 Which of the following choices is equivalent to the given expression? Teacher A B C D
148 Slide 148 / 207 Power Reducing Identities 56 Which of the following choices is equivalent to the given expression? Teacher A B C D
149 Slide 149 / 207 Power Reducing Identities 57 Which of the following choices is equivalent to the given expression? Teacher A B C D
150 Slide 150 / 207 Sum to Product Return to Table of Contents
151 Slide 151 / 207 Sum to Product Sum to Product
152 Slide 152 / 207 Sum to Product Write cos 11x + cos 9x as a product Teacher
153 Slide 153 / 207 Sum to Product Write sin 8x - sin 4x as a product Teacher
154 Slide 154 / 207 Sum to Product Find the exact value of cos 5π / 12 + cos π / 12 Teacher
155 Slide 155 / 207 Sum to Product Prove Teacher
156 Slide 156 / 207 Sum to Product Prove: Teacher
157 Slide 157 / 207 Sum to Product 58 Which of the following is equivalent to the given expression? Teacher A B C D
158 Slide 158 / 207 Sum to Product 59 Which of the following is equivalent to the given expression? Teacher A B C D
159 Slide 159 / 207 Sum to Product 60 Which of the following is not equivalent to the given expression? Teacher A B C D
160 Slide 160 / 207 Product to Sum Return to Table of Contents
161 Slide 161 / 207
162 Slide 162 / 207 Product to Sum Rewrite as a sum of trig functions. Teacher
163 Slide 163 / 207 Product to Sum Rewrite as a sum of trig functions. Teacher
164 Slide 164 / 207 Product to Sum 61 Which choice is equivalent to the expression given? Teacher A B C D
165 Slide 165 / 207
166 Slide 166 / 207 Inverse Trig Functions Return to Table of Contents
167 Slide 167 / 207
168 Slide 168 / 207
169 Slide 169 / 207 Inverse Trig Functions Inverse Trig Functions Since the cosine function does not pass the horizontal line test, we need to restrict its domain so that cos -1 is a function. cos x: Domain[0, # ] Range[-1, 1] cos -1 x: Domain[-1, 1] Range[0, π] Remember to find an inverse, switch x and y.
170 Slide 170 / 207 Inverse Trig Functions y=cos -1 x # # /2-1 1
171 Slide 171 / 207 Inverse Trig Functions Inverse Trig Functions Since the sine function does not pass the horizontal line test, we need to restrict its domain so that sin -1 is a function. sin x: Domain Range[-1, 1] sin -1 x: Domain[-1, 1] Range
172 Slide 172 / 207 Inverse Trig Functions y=sin -1 x -1 1
173 Slide 173 / 207 Inverse Trig Functions Inverse Trig Functions Since the tangent function does not pass the horizontal line test, we need to restrict its domain so that tan -1 is a function. tan x: Domain Range tan -1 x: Domain Range
174 Slide 174 / 207 Inverse Trig Functions y=tan -1 x
175 Slide 175 / 207 Inverse Trig Functions Secant
176 Slide 176 / 207 Inverse Trig Functions y=sec -1 x -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: [0, # /2) [#, 3# /2)
177 Slide 177 / 207 Inverse Trig Functions Cosecant
178 Slide 178 / 207 Inverse Trig Functions Cosecant -1 1 sec -1 x : Domain: (-#,-1] [1, # ) Range: (0, # /2] (#, 3# /2]
179 Slide 179 / 207 Inverse Trig Functions Cotangent
180 Slide 180 / 207 Inverse Trig Functions Cotangent -1 1 cot -1 x: Domain: Reals Range: (0, # )
181 Slide 181 / 207 Inverse Trig Functions Restrictions
182 Slide 182 / 207 Inverse Trig Functions Example: Evaluate the following expression. Teacher
183 Slide 183 / 207 Inverse Trig Functions Example: Evaluate the following expression. Teacher
184 Slide 184 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
185 Slide 185 / 207 Inverse Trig Functions 63 Evaluate the following expression: Teacher A B C D
186 Slide 186 / 207 Inverse Trig Functions 64 Evaluate the following expression: Teacher A B C D
187 Slide 187 / 207 Inverse Trig Functions 65 Evaluate the following expression: Teacher A B C D
188 Slide 188 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
189 Slide 189 / 207 Inverse Trig Functions Example: Evaluate the following expressions. Teacher
190 Slide 190 / 207
191 Slide 191 / 207
192 Slide 192 / 207
193 Slide 193 / 207 Trig Equations Return to Table of Contents
194 Slide 194 / 207 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Teacher Examples: Solve.
195 Slide 195 / 207 Trig Equations To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s). Teacher Examples: Solve.
196 Slide 196 / 207 Trig Equations Examples: Solve. Teacher
197 Slide 197 / 207 Trig Equations Examples: Solve. Teacher
198 Slide 198 / 207 Trig Equations 69 Find an apporoximate value of x on [0, ) that satisfies the following equation: Teacher
199 Slide 199 / 207
200 Slide 200 / 207 Trig Equations Examples: Solve. Teacher
201 Slide 201 / 207 Trig Equations Examples: Solve. Teacher
202 Slide 202 / 207 Trig Equations Examples: Solve. Teacher
203 Slide 203 / 207 Trig Equations Examples: Solve. Teacher
204 Slide 204 / 207 Trig Equations Examples: Solve. Teacher
205 Slide 205 / 207 Trig Equations Examples: Solve. Teacher
206 Slide 206 / 207 Trig Equations 71 Find an apporoximate value of x on [0, ) that satisfies the following equation: Teacher
207 Slide 207 / 207
Pre Calc. Trigonometry.
1 Pre Calc Trigonometry 2015 03 24 www.njctl.org 2 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Power Reducing
CK- 12 Algebra II with Trigonometry Concepts 1
1.1 Pythagorean Theorem and its Converse 1. 194. 6. 5 4. c = 10 5. 4 10 6. 6 5 7. Yes 8. No 9. No 10. Yes 11. No 1. No 1 1 1. ( b+ a)( a+ b) ( a + ab+ b ) 1 1 1 14. ab + c ( ab + c ) 15. Students must
2. Pythagorean Theorem:
Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle
Math Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
Math Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
CK- 12 Algebra II with Trigonometry Concepts 1
14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.
Trigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.3 Right Triangle Trigonometry Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate trigonometric
SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
Trigonometric Functions. Section 1.6
Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian
A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.
Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,
Honors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0
Chapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
CHAPTERS 5-7 TRIG. FORMULAS PACKET
CHAPTERS 5-7 TRIG. FORMULAS PACKET PRE-CALCULUS SECTION 5-2 IDENTITIES Reciprocal Identities sin x = ( 1 / csc x ) csc x = ( 1 / sin x ) cos x = ( 1 / sec x ) sec x = ( 1 / cos x ) tan x = ( 1 / cot x
Precalculus Midterm Review
Precalculus Midterm Review Date: Time: Length of exam: 2 hours Type of questions: Multiple choice (4 choices) Number of questions: 50 Format of exam: 30 questions no calculator allowed, then 20 questions
Pre-Calc Trig ~1~ NJCTL.org. Unit Circle Class Work Find the exact value of the given expression. 7. Given the terminal point ( 3, 2 10.
Unit Circle Class Work Find the exact value of the given expression. 1. cos π 3. sin 7π 3. sec π 3. tan 5π 6 5. cot 15π 6. csc 9π 7. Given the terminal point ( 3, 10 ) find tanθ 7 7 8. Given the terminal
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12)
California Mathematics Content Standards for Trigonometry (Grades 9-12) Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric
One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.
2.24 Tanz and the Reciprocals Derivatives of Other Trigonometric Functions One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the
Unit 2 - The Trigonometric Functions - Classwork
Unit 2 - The Trigonometric Functions - Classwork Given a right triangle with one of the angles named ", and the sides of the triangle relative to " named opposite, adjacent, and hypotenuse (picture on
The Other Trigonometric
The Other Trigonometric Functions By: OpenStaxCollege A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is or less, regardless
As we know, the three basic trigonometric functions are as follows: Figure 1
Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an
4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
Using the Definitions of the Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective
sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
Find all solutions cos 6. Find all solutions. 7sin 3t Find all solutions on the interval [0, 2 ) sin t 15cos t sin.
7.1 Solving Trigonometric Equations with Identities In this section, we explore the techniques needed to solve more complex trig equations: By Factoring Using the Quadratic Formula Utilizing Trig Identities
NYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order)
1 of 6 UNIT P.I. 1 - INTEGERS 1 A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable 1 A2.A.4 * Solve quadratic inequalities in one and two variables, algebraically
3.1 Fundamental Identities
www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,
1 Chapter 2 Perform arithmetic operations with polynomial expressions containing rational coefficients 2-2, 2-3, 2-4
NYS Performance Indicators Chapter Learning Objectives Text Sections Days A.N. Perform arithmetic operations with polynomial expressions containing rational coefficients. -, -5 A.A. Solve absolute value
Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach
Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t
More with Angles Reference Angles
More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o
Trigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
1 The six trigonometric functions
Spring 017 Nikos Apostolakis 1 The six trigonometric functions Given a right triangle, once we select one of its acute angles, we can describe the sides as O (opposite of ), A (adjacent to ), and H ().
1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles.
NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND OF REAL NUMBERS Name: Date: Mrs. Nguyen s Initial: LESSON 6.4 THE LAW OF SINES Review: Shortcuts to prove triangles congruent Definition of Oblique Triangles
Math Analysis Chapter 5 Notes: Analytic Trigonometric
Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot
TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
12 TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. 12.2 The Trigonometric Functions Copyright Cengage Learning. All rights reserved. The Trigonometric Functions and Their Graphs
Unit Circle. Return to. Contents
Unit Circle Return to Table of Contents 32 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant
Section 6.1 Sinusoidal Graphs
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle We noticed how the x and y values
5 Trigonometric Functions
5 Trigonometric Functions 5.1 The Unit Circle Definition 5.1 The unit circle is the circle of radius 1 centered at the origin in the xyplane: x + y = 1 Example: The point P Terminal Points (, 6 ) is on
Math 144 Activity #7 Trigonometric Identities
144 p 1 Math 144 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value
Chapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Overview: 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trig Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-sum
Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
From now on angles will be drawn with their vertex at the. The angle s initial ray will be along the positive. Think of the angle s
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 1 Chapter 8A Angles and Circles From now on angles will be drawn with their vertex at the The angle s initial ray will be along the positive.
Chapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs
5.3 Properties of Trigonometric Functions Objectives
Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.
Functions and their Graphs
Chapter One Due Monday, December 12 Functions and their Graphs Functions Domain and Range Composition and Inverses Calculator Input and Output Transformations Quadratics Functions A function yields a specific
6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
Gr. 11, 12 Pre Calculus Curriculum
LS PC. N1 Plot complex numbers using both rectangular and polar coordinates, i.e., a + bi = r(cosθ + isinθ ). Apply DeMoivre s Theorem to multiply, take roots, and raise complex numbers to a power. LS
1. Trigonometry.notebook. September 29, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
Section 5.4 The Other Trigonometric Functions
Section 5.4 The Other Trigonometric Functions Section 5.4 The Other Trigonometric Functions In the previous section, we defined the e and coe functions as ratios of the sides of a right triangle in a circle.
Section 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
The Other Trigonometric Functions
OpenStax-CNX module: m4974 The Other Trigonometric Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you
Unit 6 Trigonometric Identities
Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations Prove trigonometric identities, using: Reciprocal identities Quotient identities Pythagorean identities Sum
Chapter 5: Double-Angle and Half-Angle Identities
Haberman MTH Section II: Trigonometric Identities Chapter 5: Double-Angle and Half-Angle Identities In this chapter we will find identities that will allow us to calculate sin( ) and cos( ) if we know
CHAPTER 6. Section Two angles are supplementary. 2. Two angles are complementary if the sum of their measures is 90 radians
SECTION 6-5 CHAPTER 6 Section 6. Two angles are complementary if the sum of their measures is 90 radians. Two angles are supplementary if the sum of their measures is 80 ( radians).. A central angle of
Chapter 11B: Trig Graphing Review Sheet Test Wednesday 05/17/2017
Chapter 11B: Trig Graphing Review Sheet Test Wednesday 05/17/2017 1. The terminal ray of an angle drawn in standard position on the unit circle that measures 30 has 3 1 coordinates of,. Based on this information,
Assumption High School BELL WORK. Academic institution promoting High expectations resulting in Successful students
BELL WORK Geometry 2016 2017 Day 51 Topic: Chapter 8.3 8.4 Chapter 8 Big Ideas Measurement Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are
Chapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas
Milford Public Schools Curriculum. Department: Mathematics Course Name: Precalculus Level 1
Milford Public Schools Curriculum Department: Mathematics Course Name: Precalculus Level 1 UNIT 1 Unit Description: Students will construct polynomial graphs with zeros and end behavior, and apply limit
For a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
1.3 Basic Trigonometric Functions
www.ck1.org Chapter 1. Right Triangles and an Introduction to Trigonometry 1. Basic Trigonometric Functions Learning Objectives Find the values of the six trigonometric functions for angles in right triangles.
Summer 2017 Review For Students Entering AP Calculus AB/BC
Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus
MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically
1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram
Chapter 5 The Next Wave: MORE MODELING AND TRIGONOMETRY
ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter The Next Wave: MORE MODELING AND TRIGONOMETRY NW-1. TI-8, points; Casio, points a) An infinite number of them. b) 17p, - 7p c) Add p n to p, p
HHS Pre-Calculus Reference Book
HHS Pre-Calculus Reference Book Purpose: To create a reference book to review topics for your final exam and to prepare you for Calculus. Instructions: Students are to compose a reference book containing
List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015)
List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) MAT 155P MAT 155 1 Absolute Value Equations P 7 P 3 2 Absolute Value Inequalities P 9 P 4 3 Algebraic Expressions:
Troy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
June 9 Math 1113 sec 002 Summer 2014
June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the
Section 7.2 Addition and Subtraction Identities. In this section, we begin expanding our repertoire of trigonometric identities.
Section 7. Addition and Subtraction Identities 47 Section 7. Addition and Subtraction Identities In this section, we begin expanding our repertoire of trigonometric identities. Identities The sum and difference
4.3 Inverse Trigonometric Properties
www.ck1.org Chapter. Inverse Trigonometric Functions. Inverse Trigonometric Properties Learning Objectives Relate the concept of inverse functions to trigonometric functions. Reduce the composite function
HOW TO NOT LOSE POINTS...
Math Analysis B Final Review GROUP MATERIALS INSTRUCTIONS 1) Ms. Lee picks a student randomly. 2) Selected student chooses a question. 3) Group discusses question and writes FINAL WORK & SOLUTION on whiteboard.
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS Course Number 5121 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra
NOTES 10: ANALYTIC TRIGONOMETRY
NOTES 0: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 0. USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sin csc cos sec
WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014)
WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014) COURSE NAME: Pre-Calculus UNIT: Chapter 1 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What methods are used to solve equations
Notes on Radian Measure
MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius
and sinθ = cosb =, and we know a and b are acute angles, find cos( a+ b) Trigonometry Topics Accuplacer Review revised July 2016 sin.
Trigonometry Topics Accuplacer Revie revised July 0 You ill not be alloed to use a calculator on the Accuplacer Trigonometry test For more information, see the JCCC Testing Services ebsite at http://jcccedu/testing/
The six trigonometric functions
PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonomic Functions 4.: Trigonomic Functions of Acute Angles What you'll Learn About Right Triangle Trigonometry/ Two Famous Triangles Evaluating
A2T Trig Packet Unit 1
A2T Trig Packet Unit 1 Name: Teacher: Pd: Table of Contents Day 1: Right Triangle Trigonometry SWBAT: Solve for missing sides and angles of right triangles Pages 1-7 HW: Pages 8 and 9 in Packet Day 2:
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
nine weeks TRIGONOMETRY MAPPING # of ACT days Standard Assessment
TRIGONOMETRY MAPPING 2010-2011 1.1 Coordinate Plane Review Radicals Pythagorean Theorem Distance Formula Mid-point Formula Interval Notation Relations and Functions Vertical Line Test Content # of ACT
Benchmark Computation The student models, performs, and explains computation with complex numbers and polynomials in a variety of situations.
Standard Number and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student
Topic Outline for Algebra 2 & and Trigonometry One Year Program
Topic Outline for Algebra 2 & and Trigonometry One Year Program Algebra 2 & and Trigonometry - N - Semester 1 1. Rational Expressions 17 Days A. Factoring A2.A.7 B. Rationals A2.N.3 A2.A.17 A2.A.16 A2.A.23
MPE Review Section II: Trigonometry
MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the
The function x² + y² = 1, is the algebraic function that describes a circle with radius = 1.
8.3 The Unit Circle Outline Background Trig Function Information Unit circle Relationship between unit circle and background information 6 Trigonometric Functions Values of 6 Trig Functions The Unit Circle
ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
Chapter 4 Trigonometric Functions
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios
Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts
Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics
HS Trigonometry Mathematics CC
Course Description A pre-calculus course for the college bound student. The term includes a strong emphasis on circular and triangular trigonometric functions, graphs of trigonometric functions and identities
Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.
Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain
Pre-Calculus 40 Final Outline/Review:
2016-2017 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 16 multiple choice (32 pts) and 6 open ended (24 pts). Calculator Section: 8 multiple choice (16 pts) and 11 open ended (36 pts).
Pre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives
Pre-Calculus MATH 119 Fall 2013 Learning Objectives Section 1.1 1. Use the Distance Formula 2. Use the Midpoint Formula 4. Graph Equations Using a Graphing Utility 5. Use a Graphing Utility to Create Tables
A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019
A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.
DuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
Chapter 06: Analytic Trigonometry
Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is oneto-one. Are any of our trigonometric
I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.
Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that
Old Math 120 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 10 Exams David M. McClendon Department of Mathematics Ferris State University 1 Contents Contents Contents 1 General comments on these exams 3 Exams from Fall 016 4.1 Fall 016 Exam 1...............................
Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts
Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra