Notes for Math 112: Trigonometry
written by: Dr. Randall Paul
April 3, 2018 c March 2015 Randall Paul
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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. (a) 50◦ 770◦ = 720◦ = 2(360◦), − − − π 225◦π 5π so yes, coterminal 225◦ = = · 180 180 4 (b) 40◦ 320◦ = 360◦ = 1(360◦), ◦ ◦ − − − − so yes, coterminal (b) (c) 150◦ 440◦ = 590◦ = k(360◦), − − − 6 π 150◦π 5π so no, not coterminal 150◦ = − = − · 180◦ 180◦ − 6 (d) (c) 17π 29π 12π = = 4π = 2(2π) 3 −· 3 − 3 − − π 630◦π 7π 630◦ = = so yes, coterminal · 180◦ 180◦ 2 11 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.
s = θr (θ in radians)
Example 1.11: Find the length of the arc on a circle of radius 18 cm sub- tended by the an angle of 100◦.
s =?
100◦ 18cm
12 Example 1.12: The distance from the Earth to the Sun is approximately one hundred fifty million kilometers (1.5 108 km). Assuming a circular orbit, how far does the Earth move in four months?×
1.2.2 Angular Speed Everyone remembers the old formula for speed: distance speed = time When we talk about circular motion there are two kinds of speed: linear speed (denoted v) and angular speed (denoted ω).
arc length linear speed = v = time
angle angular speed = ω = time If we take the arc-length formula and divide both sides by time, s θ s = θr = r ⇒ t t we get the Angular Speed Formula
radians v = ωr (ω in time )
13 Example 1.13: A merry-go-round is ten meters across and spinning at a rate of 1.5 rpm (rotations per minute). What is the angular speed (in radi- ans/minute) of a child on a horse at the edge of the merry-go-round? What is the linear speed (in kilometers/hour) of the child?
Solution: The angular speed is 1.5 rpm. To put it into the appropriate units: rotations 2π radians radians ω =1.5 =3π minute · rotation minute The linear speed simply uses the Angular Speed Formula: 3π meters v = ωr = 5meters = 15π minute · minute Note ‘radians’ is a dimensionless unit and so may be dropped. We need only change linear speed to the appropriate units. meters kilometer 60 minutes kilometers v = 15π 2.83 minute · 1000 meters · hour ≈ hour
Example 1.14: What is the linear speed of the Earth (in km/hr)? (Hint: Use example 1.12)
14 1.2.3 Sector Area As well as discussing the length of an arc subtended by an angle, we may also talk about the area of the wedge subtended by an angle. This is called the Sector Area (denoted A).
If we consider the fraction of the circle swept out by the angle θ and recall the area of a circle is πr2, then we have r A θ A = θ r 2π πr2 which we solve to get the Sector Area Formula.
1 2 A = 2 θr (θ in radians)
Example 1.15: A wedge-shaped slice of pizza has an area of 60cm2. The end of the slice makes an angle of 35◦. What was the diameter of the pizza from which the slice was taken?
15 1.2.4 Practice Practice Problems (with solutions)
1. Find the length of the arc s in the fig- 5. A fan on “slow” turns at 25 rotations ure. per minute. The blades extend 18 s inches from the center.
(a) What is the angular speed of the 135 fan in rad/min? ◦ 4 m (b) What is the linear speed of the tips of the blades (in inches per minute)?
6. Two rollers are connected by a leather 2. Find the angle θ in the figure (in de- belt which is tight and does not slip. grees). The right roller is 25 cm in radius and 30 cm spinning at 6 rotations per second. The left roller is 20 cm in radius.
θ 8 cm (a) Find the angular speed of the right roller (in radians per second). (b) Find the linear speed of the belt (in cm per second). (c) Find the angular speed of the left 3. Quito, Ecuador and Libreville, Gabon roller (in radians per second). both lie on the Earth’s equator. The longitude of Quito is 78.5◦ West, while the longitude of Libreville is 9.5◦ East. (The radius of the Earth is 3960 miles.) Find the distance between the two cities. 20 25 4. Find the area of the sector shown in the figure below.
A
120◦ 6 ft
16 Homework 1.2
1. Find the length of the arc s in the fig- 5. The Greek mathematician Eratos- ure. thenes (ca. 276-195 B.C.E) measured s the radius of the Earth from the follow- ing observations. He noticed that on a certain day at noon the sun shown di- rectly down a deep well in Syene (mod- 5 m ern Aswan, Egypt). At the same time
140◦ 500 miles north on the same meridian in Alexandria the sun’s rays shown at an angle of 7.2◦ with the zenith (as mea- sured by the shadow of a vertical stick). Use this information (and the figure) to 2. Find the angle θ in the figure (in de- calculate the radius of the Earth. grees). 10 cm
Alexandria 7.2◦ Sun θ 500 5 cm Syene
3. Find the radius r of the circle in the figure. 6. Find the area of the sector shown in the figure below. 8 ft
8 m A
2 rad 80◦ 8 m r
4. Pittsburgh, PA and Miami, FL lie ap- 7. A ceiling fan with 16 inch blades rotates proximately on the same meridian (they at 45 rpm. have the same longitude). Pittsburgh (a) What is the angular speed of the has a latitude of 40.5◦ N and Miami fan (in rad/min)? 25.5◦ N. (The radius of the Earth is 3960 miles.) (b) What is the linear speed of the tips Find the distance between the two of the blades (in inches per sec- cities. ond)?
17 8. The Earth rotates about its axis once (b) Find the angular speed of the every 23 hours, 56 minutes, and 4 sec- wheel sprocket (in radians per onds. The radius of the Earth is 3960 minute). miles. (c) Find the speed of the bicycle (in What is the linear speed of a point on miles per hour). the Earth’s equator (in miles per hour)?
9. The sprockets and chain of a bicycle are shown in the figure. The pedal 13 sprocket has a radius of 5 inches, the wheel sprocket a radius of 2 inches, and 5 the wheel a radius of 13 inches. The 2 cyclist pedals at 40 rpm.
(a) Find the linear speed of the chain (in inches per minute).
Practice Solutions: 1 1 2π 1. A = θr2 = (6 ft)2 = 12π ft2 2 2 3 π 3π 135◦ = 2 · 180◦ 4 A 37.7 ft ⇒ ≈ 3π s = θr = 4m=3π m 9.4248 m 5. (a) 4 ≈ 25 rotations 2π radians 2. ω = minute · rotation 30 cm θ = =3.75 radians 8 cm radians ω = 50π ⇒ minute 180◦ 675◦ θ =3.75 = 214.9◦ · π π ≈ (b) 3. Quito is west of the Prime Meridian (0◦ 50π radians v = 18 inches Longitude), while Libreville is east, so minute · we add the longitudes. inches inches π 88◦π v = 900π 2827.4 θ = (78.5◦ +9.5◦) = ⇒ minute ≈ minute · 180◦ 180◦ θ 1.536 radians 6. (a) ⇒ ≈ d = θr (1.536)(3896 mi) 6082 mi 6 rotations 2π radians ≈ ≈ ω = R second · rotation 4. π 2π radians 120◦ = ωR = 12π · 180◦ 3 ⇒ second 18 (b) The linear speed of the belt is the (c) The linear speed of the belt is also same as the linear speed of a point the same as the linear speed of a on the right roller. point on the left roller.
12π radians v 942.5 cm/sec v = 25 cm ω = = second · L r 20 cm cm cm rad v = 300π 942.5 ω 47.1 ⇒ sec ≈ sec ⇒ L ≈ sec
19 20 Chapter 2
Trigonometric Functions
Now that we understand about angles we move on to the most important subject in this class—functions whose domain consists of angles. That is, functions which take an angle and return a real number. The ones we care about are called the trigonometric functions, and there are six of them: sine, cosine, tangent, cotangent, secant, and cosecant. In the next sections we will define these functions and discuss their properties.
2.1 Acute Angles
The trigonometric functions are defined for almost all angles from minus infinity to plus infinity. However their values are particularly easy to understand when applied to acute angles (angles between 0◦ and 90◦). Acute angles are characterized by being an interior angle of a right triangle. Say θ is an acute angle. Then θ is an interior angle in a right triangle, and we may define the six trigonometric functions as follows:
sin θ = opposite sec θ = hypotenuse opposite hypotenuse adjacent hypotenuse adjacent hypotenuse cos θ = hypotenuse csc θ = opposite θ opposite adjacent adjacent tan θ = adjacent cot θ = opposite
Example 2.1:
sin θ = 3 sec θ = 5 3 5 4 cos θ = 5 csc θ = θ tan θ = cot θ = 4 21 Some people remember the first three definitions with the acronym: SOHCAHTOA for “Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. You might fear that this definition of the trigonometric functions will depend on the size of the triangle. It does not. Below is a triangle similar to the triangle in example 2.1.
Example 2.2:
Note that still 8 4 cos θ = = 10 6 10 5 6 3 tan θ = = θ 10 4 8 etc.
We now introduce two special right triangles whose angles and sides are known exactly. The 45◦ 45◦ 90◦ Triangle: − −
π π 4 4 √2 √2 1 1 1 1 ⇒ ⇒ √2
π π 4 4 1 1 1 √2
The 30◦ 60◦ 90◦ Triangle: − −
π 6 1 1 1 √3 ⇒ √3 2 2 π 3 1 1 2 Example 2.3: Evaluate the trig functions:
√3/2 √3 sin(60◦) = 1 = 2
π tan 3 =