1 The six trigonometric functions

Size: px
Start display at page:

Download "1 The six trigonometric functions"

Transcription

1 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 (). There are six possible ways to divide one of the sides by an other side, and each of these ratios has a name: Table 1: The six trigonometric functions of Name of function Abbreviation Definition sine sin O H cosine cos tangent tan cosecant csc secant sec cotangent cot A H O A H O H A A O All these ratios are not independent of each other. If we know one of them we can figure out the rest. Example 1. The acute angle as sin. Find cos, tan, cot, sec, and csc. Answer. We can use any right triangle that has an angle with sin to calculate the other trig functions. If we have a right triangle with H, and the leg opposite to to be O, then sin O H. Such a triangle is shown below:

2 x Using the Pythagorean Theorem we can calculate: x 1 x So we have all three sides and we can calculate: cos tan cot sec csc Example. An acute angle has tan. Find the other trigonometric ratios of. Answer. The angle in the following triangle has tangent equal to. x Using the Pythagorean theorem we can calculate: x ( ) + x x So we have: sin 0 1 cos cot sec 0 csc Page

3 Exercises 1. Find the values of the other five trigonometric ratios of the acute angle given the indicated value of one of the ratios. (a) sin (b) cos (c) cos (d) sin (e) tan 9 (f) tan (g) sec 7 (h) csc Solving right triangles A triangle has three sides and three angles a total of six unknown quantities. To solve a triangle means to find the measurements of all its sides and all its angles. Example. In a right triangle ABC, with C 90, we have a units and c units. Solve this triangle. Answer. First recall the convention that when we name the corners and the sides of a triangle we use upper case letters for the corners and then we name each side with the lower case letter of the corner opposite it. So in our triangle c is the side opposite to C, and a is opposite the corner A. So we have the following: B Using the Pythagorean theorem we can calculate: b +( ) b b 1 b 1 C A So we have sinb 1 B 0 Then A 90 0 A 0 In this example we made use of the fact that we know that the sine of 0 is 1 1. What would we have done if we didn t get a sine that we recognize? Well, that s what calculators are for! All scientific calculators have buttons labeled sin 1 (as well as cos 1, and tan 1 ). These buttons give you what angle has a given sine. For example, I can use the builtin calculator in my editor to find: sin We actually cheated a bit. We assumed that if an angle has the same sine as an angle of 0, then this angle must be 0. That turns out to be true: if two acute angles have the same sine then they are equal Page

4 This means that the angle that has sine 0.8 is (approximately) Usually in the class we round to the nearest hundredth (i.e. the second decimal place), so if we need to use an angle that has sine.8 we will use.1. Example. Find in the triangle below: Answer. We have from the figure that tan Using our calculator we get: tan Rounding to the nearest hundredth, we have.. Example. In a right triangle ABC with C 90 we have that b and B 8. Solve the triangle. Answer. Such a triangle is shown below: A We have that A b C c 8 B We can calculate b using tan8 circ b. The calculator gives tan8 0.. So b 0. b.1 We also have cos8 c. The calculator gives cos So we have: c.88 c c..88 Exercises 1. In the following figure find a: a.19 1 Page

5 . In a right triangle KLM with K 90 have that M, and k. Find the lengths of l, m.. Solve the triangle ABC where A 90, B, and b.. Solve the triangle ABC where B 90, a, and b.. In a right triangle ABC with A 90 we have a and b 1. Find the angle B.. In a right triangle ABC with B 90 we have that a.1, and b. Find the angle C. 7. In a right triangle PQR we have P 90, r and q. Solve the triangle. 8. In a triangle with A 90 and cosb. find sinb. 9. For an acute angle of a right triangle we have sin. Find cos and tan.. The acute angles of a right triangle are φ and. If tan. find cosφ. Page

As we know, the three basic trigonometric functions are as follows: Figure 1

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

More information

A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.

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,

More information

Chapter 4 Trigonometric Functions

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

More information

Trigonometric Functions. Copyright Cengage Learning. All rights reserved.

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

More information

2. Pythagorean Theorem:

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

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

1.3 Basic Trigonometric Functions

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.

More information

Unit 2 - The Trigonometric Functions - Classwork

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

More information

Section 6.2 Trigonometric Functions: Unit Circle Approach

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

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles

More information

4-3 Trigonometric Functions on the Unit Circle

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

More information

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

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

More information

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach

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 information

Notes on Radian Measure

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

More information

Trigonometric Ratios of Acute Angles. Evaluate reciprocal trigonometric ratios. LEARN ABOUT the Math. In ^MNP, determine the length of MN.

Trigonometric Ratios of Acute Angles. Evaluate reciprocal trigonometric ratios. LEARN ABOUT the Math. In ^MNP, determine the length of MN. Trigonometric Ratios of cute ngles GOL Evaluate reciprocal trigonometric ratios. LERN BOUT the Math P From a position some distance away from the base of a tree, Monique uses a clinometer to determine

More information

Precalculus Midterm Review

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

More information

October 15 MATH 1113 sec. 51 Fall 2018

October 15 MATH 1113 sec. 51 Fall 2018 October 15 MATH 1113 sec. 51 Fall 2018 Section 5.5: Solving Exponential and Logarithmic Equations Base-Exponent Equality For any a > 0 with a 1, and for any real numbers x and y a x = a y if and only if

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned

More information

Exercise Set 6.2: Double-Angle and Half-Angle Formulas

Exercise Set 6.2: Double-Angle and Half-Angle Formulas Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin

More information

Using the Definitions of the Trigonometric Functions

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

More information

8-2 Trigonometric Ratios

8-2 Trigonometric Ratios 8-2 Trigonometric Ratios Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Write each fraction as a decimal rounded to the nearest hundredth. 1. 2. 0.67 0.29 Solve each equation. 3. 4. x = 7.25

More information

Solutions for Trigonometric Functions of Any Angle

Solutions for Trigonometric Functions of Any Angle Solutions for Trigonometric Functions of Any Angle I. Souldatos Answers Problem... Consider the following triangle with AB = and AC =.. Find the hypotenuse.. Find all trigonometric numbers of angle B..

More information

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 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,

More information

Practice Test - Chapter 4

Practice Test - Chapter 4 Find the value of x. Round to the nearest tenth, if necessary. Find the measure of angle θ. Round to the nearest degree, if necessary. 1. An acute angle measure and the length of the hypotenuse are given,

More information

Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :

Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the

More information

Algebra 1B. Unit 9. Algebraic Roots and Radicals. Student Reading Guide. and. Practice Problems

Algebra 1B. Unit 9. Algebraic Roots and Radicals. Student Reading Guide. and. Practice Problems Name: Date: Period: Algebra 1B Unit 9 Algebraic Roots and Radicals Student Reading Guide and Practice Problems Contents Page Number Lesson 1: Simplifying Non-Perfect Square Radicands 2 Lesson 2: Radical

More information

Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :

Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the

More information

A2T Trig Packet Unit 1

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:

More information

: SINE, COSINE, & TANGENT RATIOS

: SINE, COSINE, & TANGENT RATIOS Geometry Notes Packet Name: 9.2 9.4: SINE, COSINE, & TANGENT RATIOS Trigonometric Ratios A ratio of the lengths of two sides of a right triangle. For any acute angle, there is a leg Opposite the angle

More information

Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1.

Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1. Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean A. Definitions: 1. Geometric Mean: 2. Right Triangle Altitude Similarity Theorem: If the altitude is

More information

The Other Trigonometric

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

More information

Math 2 Trigonometry. People often use the acronym SOHCAHTOA to help remember which is which. In the triangle below: = 15

Math 2 Trigonometry. People often use the acronym SOHCAHTOA to help remember which is which. In the triangle below: = 15 Math 2 Trigonometry 1 RATIOS OF SIDES OF A RIGHT TRIANGLE Trigonometry is all about the relationships of sides of right triangles. In order to organize these relationships, each side is named in relation

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. MATH 116 Test Review sheet SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Find the complement of an angle whose measure

More information

Chapter 1. Functions 1.3. Trigonometric Functions

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

More information

Assumption High School BELL WORK. Academic institution promoting High expectations resulting in Successful students

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

More information

Name: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions

Name: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following:

More information

Trigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent

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)

More information

Algebra II B Review 5

Algebra II B Review 5 Algebra II B Review 5 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measure of the angle below. y x 40 ο a. 135º b. 50º c. 310º d. 270º Sketch

More information

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. 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

More information

Section 5.4 The Other Trigonometric Functions

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.

More information

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.

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

More information

CK- 12 Algebra II with Trigonometry Concepts 1

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.

More information

DISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK:

DISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK: Name: Class Period: DISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK: An identity is an equation that is valid for all values of the variable for which the epressions in the equation are defined. You

More information

Fundamentals of Mathematics (MATH 1510)

Fundamentals of Mathematics (MATH 1510) Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University March 14-18, 2016 Outline 1 2 s An angle AOB consists of two rays R 1 and R

More information

Pre-Calc Trigonometry

Pre-Calc Trigonometry Slide 1 / 207 Slide 2 / 207 Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double

More information

5.3 Properties of Trigonometric Functions Objectives

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.

More information

Special Angles 1 Worksheet MCR3U Jensen

Special Angles 1 Worksheet MCR3U Jensen Special Angles 1 Worksheet 1) a) Draw a right triangle that has one angle measuring 30. Label the sides using lengths 3, 2, and 1. b) Identify the adjacent and opposite sides relative to the 30 angle.

More information

3.1 Fundamental Identities

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,

More information

Transition to College Math

Transition to College Math Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain

More information

More with Angles Reference Angles

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

More information

TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.

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

More information

A List of Definitions and Theorems

A List of Definitions and Theorems Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One

More information

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations

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

More information

Calculus with business applications, Lehigh U, Lecture 05 notes Summer

Calculus with business applications, Lehigh U, Lecture 05 notes Summer Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often

More information

Unit two review (trig)

Unit two review (trig) Class: Date: Unit two review (trig) Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the reference angle for 15 in standard position? A 255 C 345

More information

Trigonometry Learning Strategies. What should students be able to do within this interactive?

Trigonometry Learning Strategies. What should students be able to do within this interactive? Trigonometry Learning Strategies What should students be able to do within this interactive? Identify a right triangle. Identify the acute reference angles. Recognize and name the sides, and hypotenuse

More information

1. Trigonometry.notebook. September 29, Trigonometry. hypotenuse opposite. Recall: adjacent

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)

More information

Pre Calc. Trigonometry.

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

More information

Chapter 5 Analytic Trigonometry

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

More information

MPE Review Section II: Trigonometry

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

More information

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of

More information

CK- 12 Algebra II with Trigonometry Concepts 1

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

More information

Essential Question How can you find a trigonometric function of an acute angle θ? opp. hyp. opp. adj. sec θ = hyp. adj.

Essential Question How can you find a trigonometric function of an acute angle θ? opp. hyp. opp. adj. sec θ = hyp. adj. . Right Triangle Trigonometry Essential Question How can you find a trigonometric function of an acute angle? Consider one of the acute angles of a right triangle. Ratios of a right triangle s side lengths

More information

AMB121F Trigonometry Notes

AMB121F Trigonometry Notes AMB11F Trigonometry Notes Trigonometry is a study of measurements of sides of triangles linked to the angles, and the application of this theory. Let ABC be right-angled so that angles A and B are acute

More information

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) 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

More information

The Other Trigonometric Functions

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

More information

Chapter 06: Analytic Trigonometry

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

More information

CHAPTERS 5-7 TRIG. FORMULAS PACKET

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

More information

Radian Measure and Angles on the Cartesian Plane

Radian Measure and Angles on the Cartesian Plane . Radian Measure and Angles on the Cartesian Plane GOAL Use the Cartesian lane to evaluate the trigonometric ratios for angles between and. LEARN ABOUT the Math Recall that the secial triangles shown can

More information

6.2 Trigonometric Integrals and Substitutions

6.2 Trigonometric Integrals and Substitutions Arkansas Tech University MATH 9: Calculus II Dr. Marcel B. Finan 6. Trigonometric Integrals and Substitutions In this section, we discuss integrals with trigonometric integrands and integrals that can

More information

Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.

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

More information

Sect 7.4 Trigonometric Functions of Any Angles

Sect 7.4 Trigonometric Functions of Any Angles Sect 7.4 Trigonometric Functions of Any Angles Objective #: Extending the definition to find the trigonometric function of any angle. Before we can extend the definition our trigonometric functions, we

More information

Jim Lambers Math 1B Fall Quarter Final Exam Solution (Version A)

Jim Lambers Math 1B Fall Quarter Final Exam Solution (Version A) Jim Lambers Math 1B Fall Quarter 004-05 Final Exam Solution (Version A) 1. Suppose that a culture initially contains 500 bacteria, and that the population doubles every hours. What is the population after

More information

JUST THE MATHS SLIDES NUMBER 3.1. TRIGONOMETRY 1 (Angles & trigonometric functions) A.J.Hobson

JUST THE MATHS SLIDES NUMBER 3.1. TRIGONOMETRY 1 (Angles & trigonometric functions) A.J.Hobson JUST THE MATHS SLIDES NUMBER 3.1 TRIGONOMETRY 1 (Angles & trigonometric functions) by A.J.Hobson 3.1.1 Introduction 3.1.2 Angular measure 3.1.3 Trigonometric functions UNIT 3.1 - TRIGONOMETRY 1 - ANGLES

More information

Practice Test - Chapter 4

Practice Test - Chapter 4 Find the value of x. Round to the nearest tenth, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite.

More information

Trigonometric Identity Practice

Trigonometric Identity Practice Trigonometric Identity Practice Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Identify the expression that completes the equation so that it is an identity.

More information

Math Analysis Chapter 5 Notes: Analytic Trigonometric

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

More information

Math 144 Activity #7 Trigonometric Identities

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

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 017/018 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and Co-Function Identities. In Section 4.6 of Chapter 4 we looked

More information

3.5 Derivatives of Trig Functions

3.5 Derivatives of Trig Functions 3.5 Derivatives of Trig Functions Problem 1 (a) Suppose we re given the right triangle below. Epress sin( ) and cos( ) in terms of the sides of the triangle. sin( ) = B C = B and cos( ) = A C = A (b) Suppose

More information

Crash Course in Trigonometry

Crash Course in Trigonometry Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................

More information

Honors Algebra 2 Chapter 14 Page 1

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

More information

( ) + ( ) ( ) ( ) Exercise Set 6.1: Sum and Difference Formulas. β =, π π. π π. β =, evaluate tan β. Simplify each of the following expressions.

( ) + ( ) ( ) ( ) Exercise Set 6.1: Sum and Difference Formulas. β =, π π. π π. β =, evaluate tan β. Simplify each of the following expressions. Simplify each of the following expressions ( x cosx + cosx ( + x ( 60 θ + ( 60 + θ 6 cos( 60 θ + cos( 60 + θ 7 cosx + cosx+ 8 x+ + x 6 6 9 ( θ 80 + ( θ + 80 0 cos( 90 + θ + cos( 90 θ 7 Given that tan (

More information

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12)

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

More information

Unit Circle. Return to. Contents

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

More information

Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72

Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72 Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72 Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree.

More information

Basic Trigonometry. DSchafer05. October 5, 2005

Basic Trigonometry. DSchafer05. October 5, 2005 Basic Trigonometry DSchafer05 October 5, 005 1 Fundementals 1.1 Trigonometric Functions There are three basic trigonometric functions, sine, cosine and tangent, whose definitions can be easily observed

More information

5-3 Solving Trigonometric Equations

5-3 Solving Trigonometric Equations Solve each equation for all values of x. 1. 5 sin x + 2 = sin x The period of sine is 2π, so you only need to find solutions on the interval. The solutions on this interval are and. Solutions on the interval

More information

Lesson 7.3 Exercises, pages

Lesson 7.3 Exercises, pages Lesson 7. Exercises, pages 8 A. Write each expression in terms of a single trigonometric function. cos u a) b) sin u cos u cot U tan U P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions 7 c)

More information

Trigonometric Functions. Section 1.6

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

More information

6.5 Trigonometric Equations

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

More information

15 x. Substitute. Multiply. Add. Find the positive square root.

15 x. Substitute. Multiply. Add. Find the positive square root. hapter Review.1 The Pythagorean Theorem (pp. 3 70) Dynamic Solutions available at igideasmath.com Find the value of. Then tell whether the side lengths form a Pythagorean triple. c 2 = a 2 + b 2 Pythagorean

More information

Chapter 5 Analytic Trigonometry

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

More information

DISTRIBUTED LEARNING

DISTRIBUTED LEARNING DISTRIBUTED LEARNING RAVEN S WNCP GRADE 12 MATHEMATICS BC Pre Calculus Math 12 Alberta Mathematics 0 1 Saskatchewan Pre Calculus Math 0 Manitoba Pre Calculus Math 40S STUDENT GUIDE AND RESOURCE BOOK The

More information

The function x² + y² = 1, is the algebraic function that describes a circle with radius = 1.

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

More information

Geometry Warm Up Right Triangles Day 8 Date

Geometry Warm Up Right Triangles Day 8 Date Geometry Warm Up Right Triangles Day 8 Name Date Questions 1 4: Use the following diagram. Round decimals to the nearest tenth. P r q Q p R 1. If PR = 12 and m R = 19, find p. 2. If m P = 58 and r = 5,

More information

Trigonometry 1st Semester Review Packet (#2) C) 3 D) 2

Trigonometry 1st Semester Review Packet (#2) C) 3 D) 2 Trigonometry 1st Semester Review Packet (#) Name Find the exact value of the trigonometric function. Do not use a calculator. 1) sec A) B) D) ) tan - 5 A) -1 B) - 1 D) - Find the indicated trigonometric

More information

PART 1: USING SCIENTIFIC CALCULATORS (50 PTS.)

PART 1: USING SCIENTIFIC CALCULATORS (50 PTS.) Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 SPRING 2018 KUNIYUKI 150 POINTS TOTAL: 50 FOR PART 1, AND 100 FOR PART 2 Show all work, simplify as appropriate,

More information

1 Chapter 2 Perform arithmetic operations with polynomial expressions containing rational coefficients 2-2, 2-3, 2-4

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

More information