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Math 150, Fall 2008, Benjamin
Aurispa
Chapter 6 Trigonometric Functions of Angles
6.1 Angle Measure
In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of
the terminal point on the unit circle. Chapter 6 will look at trig functions in terms of angles. A lot of the
information will be repetitious.
In Chapter 5, we used radians. In this Chapter, we will frequently use degrees instead of radians, although
not always.
Suppose we have an angle with initial side R1 and terminal side R2 . We can describe this angle by using
radians or by using degrees.
If we put this angle on the unit circle, the measure of the angle in radians is the length of the arc that
subtends the angle.
One degree means that the angle formed is
1
360
of a complete revolution.
180◦ = π rad and thus 360◦ = 2π.
To convert from degrees to radians, multiply by
To convert from radians to degrees, multiply by
Examples: Common angles in degrees and radians to remember:
Express 15◦ in radians.
Express
7π
6
in degrees.
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Math 150, Fall 2008, Benjamin
Aurispa
An angle is said to be in standard position if is drawn in the xy-plane with its vertex at the origin and
its angle is measured from the positive x-axis.
An angle is positive if it is measured in the counterclockwise direction, and negative if measured in the
clockwise direction.
Two angles are coterminal if their sides coincide. To obtain coterminal angles, just add/subtract 2π or
360 to the given angle.
Example: Find 2 positive and 2 negative angles that are coterminal with θ = 60◦ .
Find an angle between 0◦ and 360◦ that is coterminal with 1270◦ .
In a circle of radius r, the length s of an arc that subtends a central angle of θ in radians is
s = rθ
How do you get this formula?
Examples:
Find the length of an arc in a circle with radius 6 that subtends a central angle of
π
5.
A central angle in a circle of radius 5 is subtended by an arc of length 6 m. Find θ in radians and degrees.
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Math 150, Fall 2008, Benjamin
Aurispa
In a circle of radius r, the area A of a sector with a central angle of θ radians is
1
A = r2 θ
2
How do you get this formula?
Find the area of a sector with central angle 40◦ in a circle of with diameter 10 ft.
Example: Find the radius of a circle if the area of a sector with central angle 3 rad is 16 m2 .
There are many problems in physics that deal with circular motion, in particular linear and angular speed.
Linear speed is just the distance traveled per unit time, whereas angular speed is the rate at which the
central angle is changing.
Suppose an object moves along a circle of radius r and traverses an angle of θ radians in time t, so that the
distance traveled is s. Then angular and linear speed are given by
Angular Speed
Linear Speed
ω=
ν=
θ
t
s
t
Example: Suppose I am sitting on the edge of a merry-go-round of radius 6 ft that is making 3 revolutions
every 20 seconds. Find my angular speed and linear speed.
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Math 150, Fall 2008, Benjamin
Aurispa
Linear and angular speed along a circle of radius r are related by the formula
ν = rω
Example: A ceiling fan with 16-in blades rotates at 45 rpm (revolutions per minute).
• Find the angular speed of the fan in rad/min.
• Find the linear speed of the tips of the blades in in./min.
6.2 Trigonometry of Right Triangles
If we have a RIGHT triangle with θ as one of its acute angles, the trig functions for θ are defined as:
opposite
sin θ = hypotenuse
adjacent
cos θ = hypotenuse
csc θ = hypotenuse
opposite
sec θ = hypotenuse
adjacent
opposite
tan θ = adjacent
cot θ = adjacent
opposite
√
Example: Suppose I have a right triangle with side lengths 11, 5, and hypotenuse 6. Let θ be the angle
opposite the side of length 5. Find all 6 trig functions for θ.
√
Example: Suppose I know that tan θ =
trig functions of θ.
21
2 .
Sketch a right triangle with acute angle θ and find the other 5
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Math 150, Fall 2008, Benjamin
Aurispa
There are 2 special triangles that are used a lot. They will help you remember the trig functions for common
angles.
To solve a triangle means to determine all of its parts: the lengths of its three sides and the measures of
its three angles.
Example: I have a right triangle where I know that one of the acute angles is 60◦ and that the length of the
side opposite this angle is 10. Solve this triangle.
There are many applications that use right triangles.
Examples:
• A 20-ft ladder leans against a building so that the angle of elevation of the ladder is 72◦ . How high
does the ladder reach on the building?
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Math 150, Fall 2008, Benjamin
Aurispa
• A 60-ft wire is attached to the top of a tower, with an angle of depression of 65◦ to the ground. How
tall is the tower?
• A woman standing on a hill sees a flagpole that she knows is 60-ft tall. The angle of depression to the
bottom of the pole is 14◦ , and the angle of elevation to the top of the pole is 18◦ . Find her distance x
from the pole.
6.3 Trig Functions of Angles
In this section, we again discuss angles in the xy-plane.
Let θ be an angle in standard position and let P (x, y) be a point on the terminal side of the angle. (Note,
that we don’t say terminal point now, because P (x, y) may not be on the unit circle.) Let r be the distance
from the origin to the point P (x, y).
What is r?
Now, we have:
y
sin θ =
r
csc θ =
r
(y 6= 0)
y
x
r
tan θ =
y
(x 6= 0)
x
r
(x 6= 0)
x
cot θ =
x
(y 6= 0)
y
cos θ =
sec θ =
Note that these formulas would be exactly the same as we defined them on the unit circle when r = 1.
In Chapter 5, we talked about reference numbers. In Chapter 6, they are referred to as reference angles,
but the idea is exactly the same.
Let θ be an angle in standard position. The reference angle θ is the acute angle formed by the terminal side
of θ and the x-axis.
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Math 150, Fall 2008, Benjamin
Aurispa
Examples: Find the reference angles for the following angles θ.
θ = 820◦
θ = −105◦
The process for evaluating trig functions at any angle is very similar to before. Find the reference angle, use
common trig vaules, and determine the sign of the trig function based on the quadrant that the angle is in.
If the reference angle is not one we are familiar with, you would have to use a calculator to find the value.
Evaluate tan(870◦ )
Evaluate cot(−480◦ )
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Find all the values of the trig functions of θ given that cos θ = − 12
and θ is in Quadrant III.
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Math 150, Fall 2008, Benjamin
Aurispa
Given that cot θ = − 14 and that sin θ < 0 find the values of the other trig functions.
Recall the important identities below (the Pythagorean Identities.)
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
Write cot θ in terms of cos θ where θ is in Quadrant IV.
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1 + cot2 θ = csc2 θ