Download Chapter 13: Trigonometric Functions

Chapter Thirteen: Trigonometric Functions
Section One: Right-Triangle Trigonometry
Trigonometry is a branch of mathematics that deals mainly with triangles and special ratios between the
sides of the triangles. The most basic trigonometry deals only with right triangles. Before discussing
these ratios we will look at the names for the three sides of a right triangle. The longest side of a right
triangle is the side across from the right angle. It is called the hypotenuse. The next two sides we define
depending on which of the two acute angles we are referencing. The side that makes up the angle along
with the hypotenuse is called the adjacent side. The side that does not touch the angle is called the
opposite side.
In the figure above, angle C is the right angle.
If referencing angle A:
Side 1 is the opposite side
Side 2 is the adjacent side
Side 3 is the hypotenuse
If referencing and B:
Side 1 is the adjacent side
Side 2 is the opposite side
Side 3 is the hypotenuse
There are six trigonometric functions that describe the relationship between the 3 sides. (We will use
the Greek letter theta,  , to name our angle in these formulas.)
Sine of an angle
Cosine of an angle
opp
sin  
hyp
adj
cos 
hyp
Cosecant of an angle
hyp
csc 
opp
Secant of an angle
hyp
sec 
adj
Tangent of an angle
tan  
opp
adj
Cotangent of an angle
cot  
adj
opp
EX1: Find the values of the six trigonometric functions of X and Y in the angle below. Give the
exact and approximate answers rounded to the nearest thousandth.
We can use the six trig functions to find missing pieces of our triangle.
EX2: For the triangles below, find the missing side lengths.
a.
b.
We can use these procedures to find an angle of depression or angle of elevation (inclination).
EX3: The height of an observation tower in a state park is 30 feet. A ranger at the top of the tower sees a
fire along a line of sight that is at a 1° angle of depression. How far is the fire from the base of the
tower? Round your answer to the nearest foot.
When given the sides of a triangle and we are looking for the sides we use the inverse functions: sin 1 ,
cos 1 , and tan 1 . We sometimes call these functions arcsine, arccosine, and arctangent.
EX4: Solve the following triangles (Find all sides and angles). Keep in mind that the angles of all triangles
add up to 180 degrees.
a. fj
b.
Section Two: Angles of Rotation
This lesson focuses on angles that can be greater than 180 degrees. When a figure spins, it can rotate an
infinite number of degrees depending on how long it spins. We use the Cartestian Plane to model
rotations. All angles have a starting point (the initial side) and an ending point (the terminal side). Angles
in standard position have an initial side that lies along the positive x-axis. Positive angles spin counterclockwise and negative angles spin clockwise. A 360 degree rotation is one full spin. Therefore a 180
degree rotation is half a spin.
EX1: A ride at the amusement park makes 30 complete revolutions before stopping. How many degrees
does the ride rotate?
Thinking about an amusement park ride such as a Ferris wheel, there are multiple times throughout the
course of the ride when an individual car is in the same position. It may be in the same spot but it has
rotated more times. Angles that are in the same spot but with additional rotations are called coterminal
angles.
EX2: Find the coterminal angle  for each angle such that 360    360
a. 90
b. 210
Reference angles are extremely important in our study of trigonometry! They are simply the angle
formed between the terminal side of an angle and the nearest part of the x-axis.
EX3: Find the reference angle for the following angles
a. 120
b. 193
c. 321
d. 32
e. 405
f. 133
g. 591
h. 340
Given only a point, we can find the angle of rotation for the terminal side of an angle that would pass
through this point using the trig that we have learned thus far.
EX4: Let P  3, 5 be a point on the terminal side of  in standard position. Find the exact values of the
six trigonometric functions of  .
We will create a chart the will give us very important information concerning the signs of trig values in
the different quadrants. Take any point in each quadrant to complete the chart. Pick a second point to
check your conclusions.
Trig Value
sin and csc
cos and sec
tan and cot
I
Quadrant
II
III
IV
We can remember the signs of the trig values in the following ways. Cosine and sine are very closely
related to values of x and y. Anywhere x is positive, cosine is positive. Wherever y is negative, sine will
be negative. Tangent is similar to slope. Picturing a positive sloping line tells us that tangent is positive in
quadrants one and three. A negative slope or tangent can be seen in the second and fourth quadrants.
We can also use ACTS to determine the signs of the functions:
The image tells us which functions are positive in each quadrant: all, cos, tan, and sin.
EX5: Answer the following questions about the signs of the functions
a. If sine is positive and cosine is negative, which quadrant are you in?
b. If tangent is positive and sine is negative, which quadrant are you in?
EX6: The terminal side of  lies in quadrant IV, and cos  
5
. Find the value of all six trig functions.
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Section Three: Trigonometric Functions of Any Angle
We can use the relationships between a 45-45-90 triangle and a 30-60-90 triangle to find the trig values
of lots of common angles. From geometry, these triangles have the following relationships:
Complete the chart below using the two triangles above:
Degrees
Sine
Cosine Tangent
30
45
60
We will now discuss another very important law of trig. An angle will have the same trig values as its
reference angle.
EX1: Find the values of the following by using reference angles
a. sin(150)
b. cos(315)
c. tan( 30)
d. sin(120)
Finish filling in the chart below using the same method. You should notice a pattern.
Degrees
0
30
45
60
90
120
135
150
Sine
0
Cosine
1
1
0
Tangent
0
Degrees
180
210
225
240
undefined
270
300
315
330
360
Sine
0
Cosine
-1
Tangent
0
-1
0
undefined
EX2: Find the exact coordinates of a point, R, that is located at the intersection of a circle with a radius of
7 and the terminal side of a 135° angle in standard position.
EX3: Suppose that the length of a robotic arm is 3 meters long. Imagine that the arm starts in standard
position and rotates around a pivot point located at the origin. Find the new position of an object that is
rotated 200 degrees.
Trig functions are periodic functions. This mean that the values repeat exactly after a set amount of
values. The amount of values in the domain that it takes for the values to start repeating is called the
period of the function.
EX4: What is the period of the sine, cosine, and tangent functions?
EX5: Find the exact values of the sine, cosine, and tangent functions for each angle.
a. 1110
b. 450
c. 585
Section Four: Radian Measures and Arc Length
A radian is simply another unit in which we can measures angles. We measure radians in terms of  .
We measure radians in reference to a unit circle (a circle with a radius of 1). One radian is equal to the
angle on that circle with an arc length of one. A single radian is equal to approximately 57 . 180 is
exactly  radians. This means that a full rotation of 360 is 2 radians.
We can change between degrees and radians by multiplying by the following ratios:
Degrees to Radians Radians to Degrees

180
180

Notice that since 180 is exactly  radians, we are simply multiplying by one.
EX1: Convert the following to another form
a. 50
b.
2
radians
3
EX2: Evaluate. Give exact values.
 
sin  
4
 5 
b. cos 

 6 
 3 
c. tan 

 4 
a.
We could find the arc length of part of a circle by finding out what fractional part of the whole
circumference we are searching for and then use the whole circumference to find the small part.
EX3: Find the arc length of a sector with a radius of 3 and a central angle of

radians.
6
We can also use the following formula to find the same information: s  r , where s is the arc length, r
is the radius, and  is the central angle.
EX4: A central angle in a circle with a radius of 24 inches measures
arc intercepted by the angle.
5
radians. Find the length of the
4
When an object is spinning in a circle around a central point it has two different types of speed, linear
and angular. The linear speed is how many units of length the object travels per time (ex. Miles per
hour). The angular speed is how many degrees (or radians) the object travels per time (ex. Degrees per
second). The two are given by the following formulas:
Linear Speed:
s
r
or
t
t
Angular Speed:

t
EX5: A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4
seconds. What is the linear speed of a point on the rim of the wheel? What is the angular speed of a
point on the rim of the wheel?
Section Five: Graphing Trigonometric Functions
EX1: Use the table from section 4 to create a graph of the following functions:
a. y  sin 
b. y  cos 
c. y  tan 
We will now review the concept of transformations using the trig graphs.
The amplitude describes the height of a curve. The amplitude however is only half the height of the
graph. We find the amplitude by amplitude 
1
 max  min 
2
EX2: Describe the transformation that occurs from the parent function of each graph. What is the
amplitude and period of each graph?
a.
b.
c.
1
y  sin 
2
y  cos  3 
1 
y  2sin   
4 
We will now look at translations of trig graphs. Another name for a horizontal translation is a phase
shift.
EX3: Describe the transformation that occurs from the parent function of each graph.
a. y  sin   45
b.
y  cos  2
c.
y  cos   30 1
The tangent function translates the same way. However, these functions cannot be described using an
amplitude.
Describe each transformation from the parent function. If possible, give the amplitude and period of the
function.
a.
b.
c.
y  4sin  2   20    3
1
y  cos   15   1
2
2 
y  3tan     5
3 
Section Six: Inverses of Trigonometric Functions
We learned in section one of this chapter that if we are given an equation such as sin  
1
, then we
2
must use the inverse function of sin ( sin 1 ) to find the angle. However since we now know that trig
functions are periodic we know that this equation actually has an infinite number of answers. We can
see some of them in the graph below.
Since the inverse of a trig function is not a function, your calculator cannot give you all the possible
values. It will return only a single value for the function. However, if we can remember the period of the
function, we can find all the values of the inverse.
EX1: Find all the possible values of sin 1
3
2
The value that the calculator gives you is referred to as the function’s principal values. The chart below
gives you the principal values of each function.
sin 
90    90
cos
0    180
tan 
90    90
When we are only looking for the restricted values, we denote this by capitalizing our function. This
restriction of our domain, creates an inverse that is a function.
EX2: Evaluate the following
 1
Sin 1   
 2
 3
b. Cos 1 
 2 


a.
c.

Tan 1  3

EX3: Evaluate each expression

 3 
cos  Tan 1 
 

3



b. Cos 1  sin  315 
a.