Notes for Math 112: Trigonometry Instructor’s Version written by: Dr. Randall Paul Further problems refer to: Precalculus: Mathematics for Calculus, 4th ed. by Stewart, Redlin, and Watson December 19, 2015 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. 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Contents 1 Angles 5 1.1 DegreesandRadians ............................... 6 1.1.1 Co-terminalAngles............................ 8 1.2 RadianFormulas ................................. 11 1.2.1 Arc-length................................. 11 1.2.2 AngularSpeed .............................. 12 1.2.3 SectorArea ................................ 14 2 Trigonometric Functions 17 2.1 AcuteAngles ................................... 17 2.2 UnitCircle..................................... 22 2.2.1 ReferenceAngle.............................. 25 2.2.2 Definition of the Other Trig Functions . 27 2.2.3 PythagoreanIdentities . .. .. 27 2.3 GraphsofSineandCosine ............................ 31 2.3.1 AmplitudeandPeriod .......................... 32 2.3.2 PhaseShiftandVerticalShift . 37 2.4 OtherTrigGraphs ................................ 46 2.5 InverseTrigFunctions .............................. 53 2.5.1 InverseFunctionReview . .. .. 53 2.5.2 InverseSine ................................ 55 2.5.3 InverseCosine............................... 58 2.5.4 InverseTangent.............................. 60 2.5.5 InverseApplications ........................... 61 2.6 TrigonometricEquations ............................. 63 2.6.1 Simple Trigonometric Equations . 63 2.6.2 More Complicated Trig Equations . 66 3 Formulas 71 3.1 Identities...................................... 71 3.2 SumandDifferenceFormulas .......................... 75 3.2.1 Writing a Sum as a Single Function . 78 3.2.2 Product-to-sum and Sum-to-product Formulas . ... 81 3.2.3 DoubleAngleFormulas.......................... 83 3.2.4 HalfAngleFormulas ........................... 86 3 3.3 ComplexNumbers ................................ 91 3.3.1 TrigonometricForm............................ 91 3.3.2 Product,Quotient,andPower . 94 3.3.3 Roots ................................... 96 4 Trigonometric Geometry 101 4.1 LawofSines.................................... 101 4.1.1 AmbiguousCase ............................. 103 4.2 LawofCosines .................................. 107 4.2.1 TriangleArea ............................... 111 4.3 Vectors....................................... 113 4.3.1 TrigonometricForm............................ 115 4.3.2 Static Equilibrium . 119 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 π y = ◦ = 180◦ 4 3π Example 1.5: What is 2 in degrees? 180◦ 3π x◦ = = 270◦ π 2 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: n =0: 110◦+ 0◦ = 110◦ n =1: 110◦+ 360◦ = 470◦ n =2: 110◦+ 2 360◦ = 830◦ · n = 1: 110◦ 360◦ = 250◦ − − − n = 2: 110◦ 2 360◦ = 610◦ − − · − n = 3: 110◦ 3 360◦ = 970◦ − − · − 8 Example 1.9: Write three positive angles and three negative angles co- 7π terminal to 6 . Solution: 7π 7π n = 0 : 6 + 0= 6 7π 12π 19π n = 1 : 6 + 6 = 6 7π 24π 31π n = 2 : 6 + 6 = 6 n = 1 : 7π 12π = 5π − 6 − 6 − 6 n = 2 : 7π 24π = 17π − 6 − 6 − 6 n = 3 : 7π 36π = 29π − 6 − 6 − 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. − − What about the others? 220◦ ( 500◦) = 720◦ = 2(360◦) − − So, yes, 220◦ is coterminal to 500◦. − 600◦ ( 500◦) = 1100◦ = n(360◦) − − So, no, 600◦ is not coterminal to 500◦. − 9 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 (c) 4 (g) 8 (a) 50◦ (d) 13π (h) 23π 3 8 3π (b) 4 2. Convert the following angles measured π in degrees to angles measured in radi- (c) − 6 ans. (a) 135◦ 5. Determine whether the angles are co- (b) 400◦ terminal. (c) 250◦ − 3. Convert the following angles measured (a) 70◦ and 430◦. in radians to angles measured in de- (b) 30◦ and 330◦. grees. − 17π 5π 3π (c) 6 and 6 . (a) 8 25π (d) 32π and 11π . (b) 6 3 3 (c) 16 (approximately) − (e) 155◦ and 875◦. Further practice: Page 480 (sec 6.1) #1-40 Solutions: 3π 4π 2 3 1. (a) (b) 10 5π 3π 4 8 (c) (f) 13π 13π 3 8 (d) (g) 23π π 8 − 8 (e) (h) 2. (a) 3π (b) 20π (c) 25π 4 9 − 18 3. (a) 67.5◦ (b) 750◦ (c) 916.7◦ − 11π 19π 5π 13π 11π 23π 13π 19π 4. (a) 410◦, 770◦, 310◦, 670◦ (b) , , − , − (c) , , − , − − − 4 4 4 4 6 6 6 6 5. (a) 70◦ 430◦ = 360◦ coterminal (b)70◦ 430◦ = 360◦ coterminal (c) 17π −5π = 12π−=2π coterminal (d) 32π − 11π = 21−π =7π Not coterminal 6 − 6 6 3 − 3 3 (e)155◦ 875◦ =( 2)360◦ coterminal − − 1.2 Radian Formulas Degrees are the oldest way to measure angles, but in many ways radians are the better way to measure angles. Many formulas from calculus assume that all angles are given in radians (and this is important). The formulas below also assume angles are given in radians. 1.2.1 Arc-length In general the arc-length is the distance along a curving path. In this class we only consider the distance along a circular path. If we consider the fraction of the circle s swept out by the angle θ and recall the cir- cumference of a circle is 2πr, then we have r θ s θ = r 2π 2πr which we solve to get the Arc-length Formula.