Notes for Math 112: Trigonometry written by: Dr. Randall Paul April 3, 2018 c March 2015 Randall Paul This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The essence of the license is that You are free: to Share to copy, distribute and transmit the work • to Remix to adapt the work • Under the following conditions: Attribution You must attribute the work in the manner specified by the author (but • not in any way that suggests that they endorse you or your use of the work). Please contact the author at [email protected] to determine how best to make any attribution. Noncommercial You may not use this work for commercial purposes. • Share Alike If you alter, transform, or build upon this work, you may distribute the • resulting work only under the same or similar license to this one. With the understanding that: Waiver Any of the above conditions can be waived if you get permission from the • copyright holder. Public Domain Where the work or any of its elements is in the public domain under • applicable law, that status is in no way affected by the license. Other Rights In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights, or other applicable copyright exceptions and ⋄ limitations; The author’s moral rights; ⋄ Rights other persons may have either in the work itself or in how the work is used, ⋄ such as publicity or privacy rights. Notice For any reuse or distribution, you must make clear to others the license terms • of this work. The best way to do this is with a link to the web page below. To view a full copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Contents 1 Angles 5 1.1 DegreesandRadians ............................... 6 1.1.1 Co-terminalAngles............................ 8 1.1.2 Practice .................................. 10 1.2 RadianFormulas ................................. 12 1.2.1 Arc-length................................. 12 1.2.2 AngularSpeed .............................. 13 1.2.3 SectorArea ................................ 15 1.2.4 Practice .................................. 16 2 Trigonometric Functions 21 2.1 AcuteAngles ................................... 21 2.1.1 Practice .................................. 25 2.2 UnitCircle..................................... 28 2.2.1 ReferenceAngle.............................. 31 2.2.2 Definition of the Other Trig Functions . 33 2.2.3 PythagoreanIdentities . .. .. 33 2.2.4 Practice .................................. 36 2.3 GraphsofSineandCosine ............................ 39 2.3.1 AmplitudeandPeriod .......................... 40 2.3.2 PhaseShiftandVerticalShift . 45 2.3.3 Practice .................................. 53 2.4 OtherTrigGraphs ................................ 56 2.4.1 Practice .................................. 63 2.5 InverseTrigFunctions .............................. 64 2.5.1 InverseFunctionReview . .. .. 64 2.5.2 InverseSine ................................ 66 2.5.3 InverseCosine............................... 69 2.5.4 InverseTangent.............................. 71 2.5.5 InverseApplications ........................... 72 2.5.6 Practice .................................. 74 2.6 TrigonometricEquations ............................. 77 2.6.1 Simple Trigonometric Equations . 77 2.6.2 More Complicated Trig Equations . 80 2.6.3 Practice .................................. 84 3 3 Formulas 87 3.1 Identities...................................... 87 3.1.1 Practice .................................. 91 3.2 SumandDifferenceFormulas .......................... 92 3.2.1 Writing a Sum as a Single Function . 95 3.2.2 Practice .................................. 97 3.3 MoreTrigFormulas................................ 98 3.3.1 Product-to-sum and Sum-to-product Formulas . ... 98 3.3.2 DoubleAngleFormulas. .. .. 100 3.3.3 HalfAngleFormulas ........................... 103 3.3.4 Practice .................................. 107 3.4 ComplexNumbers ................................ 108 3.4.1 TrigonometricForm............................ 108 3.4.2 Product,Quotient,andPower . 111 3.4.3 Roots ................................... 113 3.4.4 Practice .................................. 117 4 Trigonometric Geometry 119 4.1 LawofSines.................................... 119 4.1.1 AmbiguousCase ............................. 121 4.1.2 Practice .................................. 125 4.2 LawofCosines .................................. 126 4.2.1 TriangleArea ............................... 131 4.2.2 Practice .................................. 132 4.3 Vectors....................................... 133 4.3.1 TrigonometricForm............................ 135 4.3.2 Static Equilibrium . 139 4.3.3 Practice .................................. 141 4.4 DotProduct.................................... 142 4.4.1 Practice .................................. 145 4 Chapter 1 Angles Angles measure “turning”. Counterclockwise turns are described by positive angles, and clockwise turns negative angles. Angles are described as a rotation taking one ray (called the ‘initial side’) to another ray (called the ‘terminal side’). Example 1.1: A positive angle θ. Example 1.2: A negative angle φ. terminal side initial side terminal side θ φ initial side Notice that, though they have the same initial and terminal sides, θ and φ are different angles. Definition: An angle is in standard position if its initial side is along the positive x-axis. Example 1.3: Below is the angle θ from example 1.1 in standard position. (Same rotation, different initial and terminal sides.) terminal side θ initial side 5 The x and y axes divide the plane into four quadrants. If an angle is in standard position, then its terminal side determines what quadrant the angle is in. For instance θ from example 1.3 is in quadrant II. Quadrant II Quadrant I Quadrant III Quadrant IV 1.1 Degrees and Radians There are two main units used to measure angles. In degrees a single, complete, counter- clockwise rotation is 360◦. In radians it is 2π. θ = 360◦ =2π (radians) 1 1 π Thus a positive one quarter rotation would be: 4 360◦ = 90◦ = 4 2π = 2 1 1 While a backwards one half rotation would be: 360◦ = 180◦ = 2π = π − 2 − − 2 − π 90◦ = 180◦ = π 2 − − 6 Draw the following angles in standard position. 5π 135◦ 3 (radians) 2 − When changing from degrees to radians or vice versa just remember that degrees/360 is the same fraction of a circle as radians/2π. So if an angle is x degrees and y radians, then: x y ◦ = 360◦ 2π Solving we have the formulas: 180◦ πx◦ x◦ = y and y = π 180◦ Example 1.4: What is 45◦ in radians? π45 y = ◦ = 180◦ 3π Example 1.5: What is 2 in degrees? x◦ = Example 1.6: Approximately what is 1 radian in degrees? Solution: 180◦ x◦ = 1 57.3◦ π ≈ 7 1.1.1 Co-terminal Angles Definition: Two angles are said to be co-terminal if, when in standard position, they have the same terminal side. Example 1.7: The three angles below are co-terminal. If measured in degrees then the angles θ and φ are co-terminal if and only if: θ = φ + 360◦n for some integer n. If measured in radians then the angles θ and φ are co-terminal if and only if: θ = φ +2πn for some integer n. Example 1.8: Write three positive angles and three negative angles co- terminal to 110◦. Solution: 110◦+ 0◦ = 110◦ 110◦+ 360◦ = 470◦ 110◦+ 2 360◦ = 830◦ · 110◦ 360◦ = 250◦ − − 110◦ 2 360◦ = 610◦ − · − 110◦ 3 360◦ = 970◦ − · − 8 Example 1.9: Write three positive angles and three negative angles co- 7π terminal to 6 . Notice we can also tell if two given angles are co-terminal since we know φ and θ are co-terminal if and only if φ θ = 360◦n (or 2πn if in radians). − Example 1.10: Determine which, if any, of the angles below are co-terminal. 220◦, 600◦, 500◦ − Solution: 220◦ 600◦ = 380◦ = 360◦n so 220◦ and 600◦ are not co-terminal. − − 6 What about the others? 9 1.1.2 Practice Practice Problems (with solutions) 1. Draw the following angles (the turn- 4. The measure of an angle in standard ings, not just the terminal side). position is given. Find two positive and two negative angles that are co-terminal 2π 3π 19π (a) (b) (c) to the given angle. 3 − 4 8 7π 2. Convert the following angles measured (a) 80◦ (b) − 3 in degrees to angles measured in radi- ans. 5. Determine whether the angles are co- terminal. (a) 225◦ (b) 150◦ (c) 630◦ − (a) 50 and 770 . 3. Convert the following angles measured ◦ ◦ in radians to angles measured in de- (b) 40◦ and 320◦. − grees. (c) 150◦ and 440◦. − (a) 3π (b) 7π (c) 8 (d) 17π and 29π . 4 − 6 3 3 Homework 1.1 1. Draw the following angles (the turn- 4. The measure of an angle in standard ings, not just the terminal side). position is given. Find two positive and two negative angles that are co-terminal (a) 3π (e) π 2 − 8 to the given angle. 4π 3π (b) 3 (f) 8 5π 13π 3π π (c) (g) (a) 50◦ (b) (c) 4 8 4 − 6 13π 23π (d) 3 (h) 8 5. Determine whether the angles are co- 2. Convert the following angles measured terminal. in degrees to angles measured in radi- ans. (a) 70◦ and 430◦. (a) 135◦ (b) 400◦ (c) 250◦ − (b) 30◦ and 330◦. 3. Convert the following angles measured − (c) 17π and 5π . in radians to angles measured in de- 6 6 grees. 32π 11π (d) 3 and 3 . 3π 25π (a) (b) (c) 16 (e) 155◦ and 875◦. 8 6 − 10 Practice Solutions: 3. (a) 1. (a) Divide π into thirds. 3π 180◦ 540◦π = = 135◦ 4 · π 4π (b) 2π 3 7π 180◦ 1260◦π = = 210◦ − 6 · π − 6π − (c) 180◦ 1440◦ 8 = 458.4◦ · π π ≈ (b) Divide π into quarters 4. (a) Positive: 80◦ +1 360◦ = 440◦ · 80◦ +5 360◦ = 1880◦ · Negative: 3π − 4 80◦ 1 360◦ = 280◦ − · − 80◦ 7 360◦ = 2440◦ − · − (b) Positive: (c) Divide π into eighths. 7π 6π π +1 = 19π 16 3 − 3 · 3 − 3 = + 8 8 8 Oops! Still negative. 7π 6π 5π +2 = − 3 · 3 3 7π 6π 77π + 14 = − 3 · 3 3 Negative: 19π 7π 6π π 8 +1 = − 3 · 3 − 3 7π 6π 67π 10 = − 3 − · 3 − 3 2. (a) 5.