1.1 Angles and Degree Measure
|
|
- Marybeth Fields
- 6 years ago
- Views:
Transcription
1 J. Jenkins - Math 060 Notes. Angles and Degree Measure An angle is often thought of as being formed b rotating one ra awa from a fied ra indicated b an arrow. The fied ra is the initial side and the rotated ra is the terminal side. An angle whose verte is the center of a circle is a central angle, and the arc of the circle through which the terminal side moves is the intercepted arc. An angle in standard position is located in a rectangular coordinate sstem with the verte at the origin and the initial side on the positive -ais. terminal side α initial side verte α Angle α Central angle Angle in standard position Degree Measure of Angles The measure, m, of an angle, indicates the amount of rotation to the terminal side from the initial side. It is found b using an circle centered at the verte. A arc that forms complete circle arc is 360. The degree measure of an angle is the number of degrees in the intercepted arc of a circle centered at the verte. The degree measure is positive if the rotation is counterclockwise and negative if the rotation is clockwise. - - Unit Circle
2 acute obtuse straight quadrantal The initial angle ma be rotated in a positive or negative direction to get the position of the terminal side. If the terminal side is rotated for more than one revolution, it can form a coterminal angle, which is an angle with the same terminal side. Coterminal Angles - Angles and are coterminal if and onl if there is an integer k such that m m k360. Eample : Find two positive angles and two negative angles that are coterminal with 0. Eample : Determine whether angles in standard position with the given measures are coterminal. a) 690 and 390 Eample 3: Name the quadrant in which each angle lies. a) 55 b) 650 c) 360 Minutes and Seconds Historicall, angles were measured b using the degrees-minutes-seconds format, but with calculators it is convenient to have some fractional parts of degrees written in decimal form. Each degree is divided into 60 equal parts called minutes n, and each minute is divided into 60 equal parts called seconds n. A minute is 60 of a degree and a second is 60 of a minute or of a degree Eample 4: Convert the measure 35 5 to decimal degrees. Eample 5: Convert to degrees-minutes-seconds.
3 Eample 6: Perform the indicated operations. a) b) c) Radian Measure, Arc Length, and Area - - Unit Circle The radian measure of the angle in standard position is directed length of the intercepted arc on the unit circle. For a circle with radius r, the radian measure is the length of the intercepted arc divided b r. Because the circumference of the unit circle is C r, then the circumference of the unit circle is C (r ). Conversion from degrees to radians or radians to degrees is based on the formula 80 degrees radians Eample : Convert the degree measures to radians. a) 0 b) 7. Eample : Convert the radian measures to degrees. a) 5 b) 6. 7 radians 3 3
4 - - Unit Circle Eample 3: Find two positive and two negative angles using radian measure that are coterminal with 6. Arc Length of a Circle If a 30 central angle intercepts an arc length s on a circle of radius r, then a 60 central angle intercepts an arc of length s and a 90 angle intercepts an arc length of 3s. Since a central angle of 360 intercepts an arc whose length is the circumference of the circle, we have arc length circumference m in radians radians m in degrees 360 degrees The length s of an arc intercepted b a central angle of radians on a circle of radius r is given b s r Eample 4: a) A central angle of intercepts an arc on the surface of the earth that runs from the equator to the North Pole. Using 3950 miles as the radius of the earth, find the length of the intercept arc to the nearest mile. b) The wagon wheel has a diameter of 8 inches and an angle of 30 between the spokes. What is the length of the arc s (to the nearest hundredth of an inch) between two adjacent spokes? 4
5 Eample 5: The distance between Los Angeles and New York Cit is 4505 miles. Find the central angle to the nearest tenth of a degree that intercepts an arc of 4505 miles on the surface of the earth (radius 3950 miles). Area of a Sector of a Circle area of sector m in radians area of circle radians m in degrees 360 degrees Area of a Sector: The area A of a sector with central angle (in radians) in a circle of radius r is given b A r Eample 6: A center-pivot irrigation sstem is used to water a circular field with radius 00 feet. In three hours the sstem waters a sector with a central angle of. What area (in 8 square feet) is watered at that time?.3 Angular and Linear Velocit If the speedometer on our car sas 50 mph, then our velocit is 50 mph. Velocit is the rate at which the location of an object is changing with respect to time. We will discuss two tpes of velocit. The angular velocit of a point is the rate at which the angle is changing. The linear velocit of a point in motion is the rate at which the distance is changing. Angular Velocit 400 revolutions per minute could be considered angular velocit, however we will epress angular velocit in radians per unit of time. In order to convert from revolutions to radians, we use the fact that rad rev. 400 rev 400 rev rad 800 rad 53 rad/ min min min rev min We use the greek letter (omega) to represent angular velocit. Definition: If a point is in motion on a circle through an angle of radians in time t, then its angular velocit is given b t Eample : Changing the units Convert the angular velocit of 3000 rad/hr to rad/ sec. 5
6 Eample : Finding angular velocit A tpical lawnmower blade rotates at a rate of 350 revolutions per minute (rpm). What is the angular velocit in radians per second of a point on the tip of the blade? Linear Velocit If a point is in motion on a circle then the distance traveled b the point in a unit of time is the length of an arc on the circle. We use the letter v to represent linear velocit for a point in circular motion. Definition: If a point is in motion on a circle of radius r through an angle of radians in time t, then its linear velocit is given b v s t r t Eample 3: Linear Velocit of a propeller A propeller with a radius of 8 meters is rotating at 6 revolutions per second. What is the linear velocit in meters per second for a point on the tip of the propeller? Eample 4: What is the linear velocit in miles per hour of the tip of a 4 inch lawnmower blade that is rotating at 000 revolutions per minute? Linear Velocit in Terms of Angular Velocit Linear velocit is arc length over time v s/t and angular velocit is angle over time. Since s r, we have t v s t r r t r t We have found the relationship between the two velocities. Theorem: If v is the linear velocit of a point on a circle of radius r, and is its angular velocit, then v r An point on the surface of the earth (eept at the poles) makes one revolution radians about the ais of the earth is 4 hours. So the angular velocit of a point on the earth is 4 radians per hour. The linear velocit of a point on the surface of the earth depends on or its distance for the ais of the earth. Eample 5: Linear velocit on the surface of the earth What is the linear velocit in feet per second for a point on the equator? 6
7 .4 The Trigonometric Functions The si trigonometric functions are the sine, cosine, tangent, cosecant, secant, and cotangent functions. Abbreviations for the functions are sin, cos, tan, csc, sec, and cot, respectivel. There are several was to define these functions of trig. Definition of the Trigonometric Functions Suppose that is an angle in standard position whose terminal side contains the point,. Let r be the distance between, and the origin. Using the distance formula, r. The trigonmetric ratios are the si possible ratios that can be formed with the numbers,, and r, provided that no denominator is zero. (,) *Remember SOH CAH TOA r α The Trigonometric Fuctions sin opp hp r csc hp opp r cos adj hp r sec hp tan opp adj cot opp adj adj r Note that csc, cos, and tan are reciprocals for sin, cos, and tan, respectivel. Reciprocal Identities csc sec sin cos cot tan Eample : Evaluating the trig functions Find the values of the si trigonometric functions of the angle in standard position whose terminal side passes through the point,4. To evaluate trig functions for an angle, we must have a point, on the terminal side of the angle. The trig functions have the same values regardless of which point is used. If, and, are two points on the terminal side, then the two triangles formed are similar. Because similar triangles are proportional, we get the same values for the trig functions. 7
8 (',') r' (,) r ' α ' e) sin r r Eample : Evaluating trig functions for quadrantal angles Find the eact values. a) csc90 b) tan80 c) sin 3 d) cot0 Quadrants Find the signs of each function in the quadrants A good mnemonic for the signs of the basic function is "All Students Take Calculus" Trigonometric Functions at Multiples of triangle sin 45 cos45 tan 45 csc 45 sec 45 cot 45 8
9 Eample 3: Find the eact value of each function. a) sin 3 b) tan c) cos Trigonometric Functions at Multiples of 30 A triangle is made b making two congruent triangles formed b the altitude of an equilateral triangle
10 Eample 4: Find the eact value of each function a) sin60 b) cos 7 6 c) tan Eample 5: Find each function value rounded to four decimal places. a) sin3. 5 b) cos85. 3 c) tan 9 d) csc4. 5 e) sec3. 5 f) cot49. 0
11 .5 Right Triangle Trigonometr Inverse Trigonometric Functions A solution to the equation sin is an angle whose sine is. Because sin 30 and sin 50, could be 30 or 50. Since an angle with the same terminal side as 30 or 50 is a solution, there are infinitel man solutions. (Since right triangles other two angles are onl acute angles, we are onl interested in acute angle measures). Eample : Find the angle that satisfies each equation where a) cos b) sin c) tan In order to solve for algebraicall, take the inverse of both sides of the equation. Taking the inverse works for sin, cos, and tan equations. 3 e) sin sin sin sin 3 60 Definition of Inverse Trigonometric Functions sin provided sin and cos provided cos and 0 80 tan provided tan and Eample : Evaluate each epression. Give the result in degrees. a) sin 3 b) cos c) tan 3 Eample 3: Evaluate each epression. Give the result in degrees to the nearest tenth. a) sin 3/7 b) cos c) tan 3. 4 Right Triangles The trigonometric fuctions of an acute angle of a right triangle sin opp hp r csc hp opp r r cos adj hp r sec hp adj r tan opp adj cot opp adj
12 Eample 4: Find all si trigonometric functions for the angle. α c 5 Eample 5: Solve the right triangle (find all missing sides and angles) β 60 b a Eample 6: Solve the right triangle given a and b 5. Find the acute angles to the nearest tenth of a degree. Applications Using trigonometr, we can find the size of an object without actuall measuing the object. Two common terms used in regard are angle of elevation and angle of depression. β angle of depression α angle of elevation horizontal line
13 Eample 7: The angle of elevation of the top of a cell phone tower is 38. at a distance of 344 feet from the tower. What ist he height of the tower? Eample 8: At one location, the angle of elevation of the top of an antenna is 44.. At a point that is 00 feet closer to the antenna, the angle of elevation is 63.. What is the height of the antenna? Eample 9: A sailor spots a small island on the horizon from the top of a 40 ft mast. How far is it to the island? Use 3950 miles for the radius of the earth. Eample 0: Graphing Calculator Required It is about 00 miles from Houston to San Antonio on interstate I-0. A pilot at an altitude of 5 miles over Houston spots San Antonio on the horizon. From this information calculate the radius of the earth. 3
14 .6 The Fundamental Identit and Reference Angles The fundamental identi involves the squares of sine and cosine functions. For convenience, we write sin as sin and cos as cos. B definition sin r, cos r, and r. So sin cos r 3 r r r r Fundamental Identit of Trigonometr: sin cos Eample : Find sin if cos 4 and is an angle in quadrant IV. Reference Angle: If is a nonquadrantal angle in standard position, then the reference angle for is the positive acute angle (reads "theta prime") formed b the terminal side of and the positive or negative -asis. θ'= π - θ θ θ' θ' θ θ θ'= θ - π θ θ'= π - θ θ' θ' 4
15 Eample : For each angle, sketch the reference angle and give the measure of in both radians and degrees. a) 50 b) 5 c) 4 7 d) Eample 3: For each angle, find the sine and cosine using the reference angle. a) 50 b) 5 c) 4 7 d) Modeling with the Sine Function The trigonometric functions can be used to model periodic phenomena. Eample 4: The tide in the Ba of Fund rises and falls ever 3 hours. The depth of the water at a certain point in the ba is modeled b the function d 8 sin t 9, where t is 3 time in hours and d is depth in meters. Find the depth at t 3 (high tide) and t (low tide)
16 Chapter Review. Find a coterminal angle in radians or degrees m m k360 or m m k. Convert from decimal degrees to degrees, minutes, seconds Convert from degrees, minutes, seconds to decimal degrees minute 60 degree second 60 minute 3600 degree 3. Add, subtract, multipl, and divide degrees, minutes, seconds. 4. Convert from radians to degrees and degrees to radians. 80 degrees radians 5. Know and understand how to use arc length circumference 6. Find the arc length s r given that is in radians. m in radians radians m in degrees 360 degrees 7. Find the area of a sector A r 8. Convert from revolutions to radians or degrees rad 360 rev 9. Use conversions to change units (meters, inches, feet, miles, seconds, minutes, hours, etc.) 0. Find angular velocit t. Find linear velocit v s t r r t. The Trigonometric Fuctions sin opp hp r cos adj hp r csc opp hp r sec hp adj r r tan opp adj cot opp adj 3. Reciprocal Identities csc sec sin cos cot tan 4. Evaluate eact values of sin, cos, tan, csc, sec, and cot using reference angles. 5. Evaluate decimal values of sin, cos, tan, csc, sec, and cot using a calculator. 6. Know and understand the special right triangles and Know and understand the Unit Circle ( degrees, radians, ordered pairs) 8. Find an angle using trig functions, the unit circle, inverse functions, and a calculator. 9. Know and understand the signs of the trig functions in each quadrant (ASTC). 0. Solve a right triangle (find all missing sides and angles). Solve problems using angle of elevation and angle of depression.. Know and understand the fundamental identit of trigonometr sin cos 3. Find functions using a reference angle and the correct quadrant signs. 4. Know how to solve problems given formulas of a periodic phenomena. (sine wave) 6
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:.
More information5.1 Angles and Their Measurements
Graduate T.A. Department of Mathematics Dnamical Sstems and Chaos San Diego State Universit November 8, 2011 A ra is the set of points which are part of a line which is finite in one direction, but infinite
More informationCHAPTER 4 Trigonometry
CHAPTER Trigonometr Section. Radian and Degree Measure You should know the following basic facts about angles, their measurement, and their applications. Tpes of Angles: (a) Acute: Measure between 0 and
More informationMath 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 information2018 Midterm Review Trigonometry: Midterm Review A Missive from the Math Department Trigonometry Work Problems Study For Understanding Read Actively
Summer . Fill in the blank to correctl complete the sentence..4 written in degrees and minutes is..4 written in degrees and minutes is.. Find the complement and the supplement of the given angle. The complement
More informationFundamentals 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 informationAn can be formed by rotating one ray away from a fixed ray indicated by an arrow. The fixed. ray is the and the rotated ray is the.
Date: 1/29/19 61 Section: Objective: angle angles t their measures An can be formed by rotating one ray away from a fixed ray indicated by an arrow The fixed initial side terminal side ray is the and the
More informationChapter 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
More information1.1 Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 1) 162
Math 00 Midterm Review Dugopolski Trigonometr Edition, Chapter and. Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. ) ) - ) For the given angle,
More informationTransition 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 informationn power Name: NOTES 2.5, Date: Period: Mrs. Nguyen s Initial: LESSON 2.5 MODELING VARIATION
NOTES 2.5, 6.1 6.3 Name: Date: Period: Mrs. Nguyen s Initial: LESSON 2.5 MODELING VARIATION Direct Variation y mx b when b 0 or y mx or y kx y kx and k 0 - y varies directly as x - y is directly proportional
More information1. 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the appropriate identity to find the indicated function value. Rationalize the denominator,
More informationMath 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 informationPrecalculus Lesson 6.1: Angles and Their Measure Lesson 6.2: A Unit Circle Approach Part 2
Precalculus Lesson 6.1: Angles and Their Measure Lesson 6.2: A Unit Circle Approach Part 2 Lesson 6.2 Before we look at the unit circle with respect to the trigonometric functions, we need to get some
More informationPractice 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 informationDISTRIBUTED 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 informationAngles and Applications
CHAPTER 1 Angles and Applications 1.1 Introduction Trigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and angles of a triangle. Plane trigonometry, which is the
More informationSection 6.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.
1 Section 6.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint
More informationChapter 1: Trigonometric Functions 1. Find (a) the complement and (b) the supplement of 61. Show all work and / or support your answer.
Trig Exam Review F07 O Brien Trigonometry Exam Review: Chapters,, To adequately prepare for the exam, try to work these review problems using only the trigonometry knowledge which you have internalized
More informationChapter 13: Trigonometry Unit 1
Chapter 13: Trigonometry Unit 1 Lesson 1: Radian Measure Lesson 2: Coterminal Angles Lesson 3: Reference Angles Lesson 4: The Unit Circle Lesson 5: Trig Exact Values Lesson 6: Trig Exact Values, Radian
More informationA 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 informationTrigonometry.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 informationMth 133 Trigonometry Review Problems for the Final Examination
Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More informationc arc length radius a r radians degrees The proportion can be used to
Advanced Functions Page of Radian Measures Angles can be measured using degrees or radians. Radian is the measure of an angle. It is defined as the angle subtended at the centre of the circle in the ratio
More informationA Short Course in Basic Trigonometry. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A Short Course in Basic Trigonometry Marcel B. Finan Arkansas Tech University c All Rights Reserved PREFACE Trigonometry in modern time is an indispensable tool in Physics, engineering, computer science,
More informationAn angle on the coordinate plane is in standard form when the vertex is on the origin and one ray lies on the positive x-axis.
Name: Topic: Main Ideas/Questions Notes/Eamples Date: Class: Angles in Standard Form y θ An angle on the coordinate plane is in standard form when the verte is on the origin and one ray lies on the positive
More informationFind the length of an arc that subtends a central angle of 45 in a circle of radius 8 m. Round your answer to 3 decimal places.
Chapter 6 Practice Test Find the radian measure of the angle with the given degree measure. (Round your answer to three decimal places.) 80 Find the degree measure of the angle with the given radian measure:
More informationPractice 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 informationCHAPTER 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
More informationUnit 6 Introduction to Trigonometry Degrees and Radians (Unit 6.2)
Unit 6 Introduction to Trigonometr Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Understand an angle
More informationMath 1303 Part II. The opening of one of 360 equal central angles of a circle is what we chose to represent 1 degree
Math 1303 Part II We have discussed two ways of measuring angles; degrees and radians The opening of one of 360 equal central angles of a circle is what we chose to represent 1 degree We defined a radian
More information(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER
PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places
More informationMAC 1114: Trigonometry Notes
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
More informationTrigonometry 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 informationI IV II III 4.1 RADIAN AND DEGREE MEASURES (DAY ONE) COMPLEMENTARY angles add to90 SUPPLEMENTARY angles add to 180
4.1 RADIAN AND DEGREE MEASURES (DAY ONE) TRIGONOMETRY: the study of the relationship between the angles and sides of a triangle from the Greek word for triangle ( trigonon) (trigonon ) and measure ( metria)
More information1.1 Angles, Degrees, and Arcs
MA140 Trig 2015 Homework p. 1 Name: 1.1 Angles, Degrees, and Arcs Find the fraction of a counterclockwise revolution that will form an angle with the indicated number of degrees. 3(a). 45 3(b). 150 3(c).
More informationSection 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 informationMore 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( 3 ) = (r) cos (390 ) =
MATH 7A Test 4 SAMPLE This test is in two parts. On part one, you may not use a calculator; on part two, a (non-graphing) calculator is necessary. When you complete part one, you turn it in and get part
More informationCHAPTER 1. ANGLES AND BASIC TRIG
DR. YOU: 017 FALL 1 CHAPTER 1. ANGLES AND BASIC TRIG LECTURE 1-0 REVIEW EXAMPLE 1 YOUR TURN 1 Simplify the radical expression. Simplify the radical expression. (A) 108 (A) 50 First, find the biggest perfect
More informationMIDTERM 3 SOLUTIONS (CHAPTER 4) INTRODUCTION TO TRIGONOMETRY; MATH 141 SPRING 2018 KUNIYUKI 150 POINTS TOTAL: 30 FOR PART 1, AND 120 FOR PART 2
MIDTERM SOLUTIONS (CHAPTER 4) INTRODUCTION TO TRIGONOMETRY; MATH 4 SPRING 08 KUNIYUKI 50 POINTS TOTAL: 0 FOR PART, AND 0 FOR PART PART : USING SCIENTIFIC CALCULATORS (0 PTS.) ( ) = 0., where 0 θ < 0. Give
More information5.3 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) ±
Final Review for Pre Calculus 009 Semester Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation algebraically. ) v + 5 = 7 - v
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationExercise Set 4.3: Unit Circle Trigonometry
Eercise Set.: Unit Circle Trigonometr Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle. Sketch each of the following angles in
More information4-3 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 information1) 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
More informationTrigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.1 Radian and Degree Measure Copyright Cengage Learning. All rights reserved. What You Should Learn Describe angles. Use radian
More informationTrigonometry 1 Review for the District Final
Review for the District Final Directions: There are 4 multiple-choice questions (1-4). Do not write in this test booklet. Read each question carefully. Fill in the circle (A, B, C, or D) for the best answer
More informationAlgebra II Standard Term 4 Review packet Test will be 60 Minutes 50 Questions
Algebra II Standard Term Review packet 2017 NAME Test will be 0 Minutes 0 Questions DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.
More informationName Date Period. Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PreAP Precalculus Spring Final Exam Review Name Date Period Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression.
More informationTrigonometric Functions
Trigonometric Functions 015 College Board. All rights reserved. Unit Overview In this unit ou will build on our understanding of right triangle trigonometr as ou stud angles in radian measure, trigonometric
More informationA2T 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 informationChapter 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(A) (12, 5) (B) ( 8, 15) (C) (3,6) (D) (4,4)
DR. YOU: 018 FALL 1 CHAPTER 1. ANGLES AND BASIC TRIG LECTURE 1-0 REVIEW EXAMPLE 1 YOUR TURN 1 Simplify the radical expression. Simplify the radical expression. (A) 108 (A) 50 First, find the biggest perfect
More informationChapter 3. Radian Measure and Circular Functions. Section 3.1: Radian Measure. π 1.57, 1 is the only integer value in the
Chapter Radian Measure and Circular Functions Section.: Radian Measure. Since θ is in quadrant I, 0 < θ
More information: 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 informationTrigonometric 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 informationSHORT 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 informationSection 4.2: Radians, Arc Length, and the Area of a Sector
CHAPTER 4 Trigonometric Functions Section 4.: Radians, Arc Length, and the Area of a Sector Measure of an Angle Formulas for Arc Length and Sector Area Measure of an Angle Degree Measure: 368 SECTION 4.
More informationHonors Accelerated Pre-Calculus Midterm Exam Review Name: January 2010 Chapter 1: Functions and Their Graphs
Honors Accelerated Pre-Calculus Midterm Eam Review Name: January 010 Chapter 1: Functions and Their Graphs 1. Evaluate the function at each specified value of the independent variable and simplify. 1 f
More informationChapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians = 180º Therefore, o 180 π 1 rad =, or π 1º = 180 rad Angle Measure Conversions π 1. To convert degrees to radians,
More informationMath 175: Chapter 6 Review: Trigonometric Functions
Math 175: Chapter 6 Review: Trigonometric Functions In order to prepare for a test on Chapter 6, you need to understand and be able to work problems involving the following topics. A. Can you sketch an
More informationNotes 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 informationPractice Questions for Midterm 2 - Math 1060Q - Fall 2013
Eam Review Practice Questions for Midterm - Math 060Q - Fall 0 The following is a selection of problems to help prepare ou for the second midterm eam. Please note the following: anthing from Module/Chapter
More informationTrigonometry IN CAREERS. There are many careers that use trigonometry. Several are listed below.
Trigonometr. Radian and Degree Measure. Trigonometric Functions: The Unit Circle. Right Triangle Trigonometr. Trigonometric Functions of An Angle.5 Graphs of Sine and Cosine Functions.6 Graphs of Other
More information150 Lecture Notes - Section 6.1 Angle Measure
c Marcia Drost, February, 008 Definition of Terms 50 Lecture Notes - Section 6. Angle Measure ray a line angle vertex two rays with a common endpoint the common endpoint initial side terminal side Standard
More informationLesson 10.2 Radian Measure and Arc Length
Lesson 10.1 Defining the Circular Functions 1. Find the eact value of each epression. a. sin 0 b. cos 5 c. sin 150 d. cos 5 e. sin(0 ) f. sin(10 ) g. sin 15 h. cos 0 i. sin(0 ) j. sin 90 k. sin 70 l. sin
More informationMATH 130 FINAL REVIEW
MATH 130 FINAL REVIEW Problems 1 5 refer to triangle ABC, with C=90º. Solve for the missing information. 1. A = 40, c = 36m. B = 53 30', b = 75mm 3. a = 91 ft, b = 85 ft 4. B = 1, c = 4. ft 5. A = 66 54',
More informationSection 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 information5.1: Angles and Radian Measure Date: Pre-Calculus
5.1: Angles and Radian Measure Date: Pre-Calculus *Use Section 5.1 (beginning on pg. 482) to complete the following Trigonometry: measurement of triangles An angle is formed by two rays that have a common
More informationCh6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2
Ch6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2 2. Find the degree measure of the angle with the given radian measure.
More informationSection 6.1 Angles and Radian Measure Review If you measured the distance around a circle in terms of its radius, what is the unit of measure?
Section 6.1 Angles and Radian Measure Review If you measured the distance around a circle in terms of its radius, what is the unit of measure? In relationship to a circle, if I go half way around the edge
More informationExercise 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 informationUnit 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 informationMATH 1316 REVIEW FOR FINAL EXAM
MATH 116 REVIEW FOR FINAL EXAM Problem Answer 1. Find the complete solution (to the nearest tenth) if 4.5, 4.9 sinθ-.9854497 and 0 θ < π.. Solve sin θ 0, if 0 θ < π. π π,. How many solutions does cos θ
More informationCollege Trigonometry
College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 11 George Voutsadakis (LSSU) Trigonometry January 015 1 / 8 Outline 1 Trigonometric
More informationAlgebra/Pre-calc Review
Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge
More informationUnit 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 informationJUST 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 informationChapter 3. Radian Measure and Circular Functions. Copyright 2005 Pearson Education, Inc.
Chapter 3 Radian Measure and Circular Functions Copyright 2005 Pearson Education, Inc. 3.1 Radian Measure Copyright 2005 Pearson Education, Inc. Measuring Angles Thus far we have measured angles in degrees
More informationCK- 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 information3 a = b = Period: a = b = Period: Phase Shift: V. Shift: Phase shift: V. Shift:
Name: Semester One Eam Review Pre-Calculus I. Second Nine Weeks Graphing Trig Functions: sketch the graph of the function, identif the parts being asked. 1. sin. cos( ) 1 Domain: Range: Domain: Range:
More informationGroup/In-Class Exercises 8/18/09 g0401larson8etrig.tst 4.1 Radian and Degree Measure
Group/In-Class Exercises 8/8/09 g040larson8etrig.tst 4. Radian and Degree Measure Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The given angle
More informationEssential 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 informationAPPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I
APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as
More informationREVIEW, pages
REVIEW, pages 5 5.. Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. a) tan (5 ) b) cos c) sec ( ) cos º cos ( ) cos
More informationBRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n + n ( 9 )( ) + + 6 + 9ab
More informationModule 2: Trigonometry
Principles of Mathematics 1 Contents 1 Module : Trigonometr Section 1 Trigonometric Functions 3 Lesson 1 The Trigonometric Values for θ, 0 θ 360 5 Lesson Solving Trigonometric Equations, 0 θ 360 9 Lesson
More informationGiven one trigonometric ratio and quadrant, determining the remaining function values
MATH 2412 Precalculus Sections 4.1-4.5 Trigonometry (quick review) Below is a list of topics you should be familiar with if you have completed a course recently in Trigonometry. I am going to assume knowledge
More informationChapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians 180º Therefore, o 180 π 1 rad, or π 1º 180 rad Angle Measure Conversions π 1. To convert degrees to radians, multiply
More informationBRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n + n ( 9 ) ( ) + + 6 + 9ab
More informationMPE 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; approximate b to the nearest tenth and B or β to the nearest minute. Hint: Draw a triangle. B = = B. b cos 49.7 = 215.
M 1500 am Summer 009 1) Given with 90, c 15.1, and α 9 ; approimate b to the nearest tenth and or β to the nearest minute. Hint: raw a triangle. b 18., 0 18 90 9 0 18 b 19.9, 0 58 b b 1.0, 0 18 cos 9.7
More information4 Trigonometric Functions. Chapter Contents. Trigonometric Equations. Angles and Their Measurement. The Sine and Cosine Functions
9788405637_CH04_st.qd 0/5/4 5:33 PM Page 06 Jones & Bartlett Learning.. gualtiero boffi/shutterstock, Inc. 4 Trigonometric Functions Chapter Contents 4. Angles and Their Measurement 4.8 Trigonometric Equations
More informationMath 122 Final Review Guide
Math 122 Final Review Guide Some questions are a combination of ideas from more than one section. Listed are the main sections that a question relates to. 5.4 1. Convert 97 to radians. 5.3 2. If 1035 is
More information1. Evaluate the function at each specified value of the independent variable and simplify. f 2a.)
Honors Pre-Calculus Midterm Eam Review Name: January 04 Chapter : Functions and Their Graphs. Evaluate the function at each specified value of the independent variable and simplify. f ( ) f () b. f ( )
More informationPractice Questions for Midterm 2 - Math 1060Q Fall
Eam Review Practice Questions for Midterm - Math 00Q - 0Fall The following is a selection of problems to help prepare ou for the second midterm eam. Please note the following: there ma be mistakes the
More information