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Trigonometry
Angles & Circular Functions: Circular Functions
Defining Sine and Cosine
The trig functions are sine, cosine, tangent, cotangent, secant, & cosecant. Trig functions are
evaluated at angles. For example, the sine of a 30° angle equals 0.5. The value of a trig
function at an angle is based on what the sides of a triangle would be if that were the angle in the
triangle. To explore these functions, we will start with sine and cosine.
To define sine and cosine, we will place the angle, θ , in standard position on an x-y coordinate
plane. Pick any point on the terminal side and label it (x, y).
For any angle,θ , in standard position with a point (x, y) on its terminal
side, the distance from the origin to the point (x, y) being r =
the sine and cosine are
x
y
cos θ = and sin θ = .
r
r
Example:
x 2 + y2 ,
Find the sine and cosine of an angle,θ , in standard position if the point (3, 4) lies
on its terminal side. Sketch the angle.
Calculate r:
r=
6
x 2 + y2
(3, 4)
4
r = 9 + 16
r=5
r
2
For the angle in this
position, x = 3, y = 4
and -10
r = 5.
x 3
=
r 5
y 4
sinθ = =
r 5
cosθ =
θ
-5
(3, 0)
-2
-4
5
Example 2:
8
and the terminal side of the angle θ is in the first
17
quadrant. Sketch the angle:
Find sin θ if cos θ =
Since cos θ =
20
8
, then x = 8 & r = 17.
17
Find y.
r2 = x 2 + y 2
15
289 = 64 + y 2
225 = y 2
±15 = y
10
17
y
5
Since we were told the
angle was in Q1, y must
be positive 15.
θ
-10
8
-5
-10
-15
10
20
30
Now we know, x = 8,
y = 15 & r = 17, so
sinθ =
15
17
The Unit Circle
Since there are infinite points, (x, y), on the terminal side of the angle,θ , we may as well choose
the easiest point to use in our calculations. We will use the points that have r = 1. Every angle
has a point at this radius. These points form the Unit Circle (the radius of the circle is one unit).
With r = 1, the definitions of sine and cosine change.
If the terminal side of an angle θ in standard position intersects the unit circle at a point (x , y ),
then
cos θ = x and sin θ = y .
Below is drawn a circle with radius 1. An angle,θ , is shown on the plane. Since r = 1, the point
where the terminal side intersects the circle, (x, y) can be translated into (cos θ , sin θ ) .
1
P (cosθ , sin θ )
θ
-2
1
2
-1
While it may be a challenge to write the ordered pairs for most of the points on this circle, there
are four points that we can label easily.
2
(0,1)
The Unit Circle
1
(-1,0)
(1,0)
-2
-1
(0,-1)
Use the diagram above to calculate the following.
1. Find the sine of 90° (abbreviated sin 90°).
Imagine a 90° angle in standard position. Where does the terminal side land?
What is the ordered pair at the end of the terminal side? The sine of 90° is the
y-value of that ordered pair!
sin 90˚ = 1
2. Find cos π:
Imagine a π (radians) angle in standard position. Where does the terminal
side land? What is the ordered pair at the end of the terminal side? The
cosine of π is the x-value of that ordered pair!
cos π = - 1
3. You try these:
a. cos 270˚ = _________
π
b. cos
= _________
2
c. sin 4π = _________
9π
= _________
d. sin
2
The Six Trigonometric Functions
The trig functions are sine, cosine, tangent, cotangent, secant, & cosecant. To define all six trig
functions, we will place the angle,θ , in standard position on an x-y coordinate plane. Pick any
point on the terminal side and label it (x, y).
For any angle,θ , in standard position with a point (x, y) on its terminal side, the
x 2 + y2 , the trig functions
distance from the origin to the point (x, y) being r =
are defined as the following:
y
r
x
cos θ =
r
y
tan θ =
x
sinθ =
Example:
r
y
r
secθ =
x
x
cot θ =
y
csc θ =
The terminal side of an angle, θ , in standard position contains the point (8, -15).
Evaluate all six trigonometric functions for this angle.
First, find r. (Note, even though ‘y’ is negative, ‘r’ is always positive.)
r = x 2 + y2
r = 64 + 225
8
r = 289
r = 17
10
-5
x = 8, y = -15, r = 17
r
- 15
-10
15
17
8
cosθ =
17
15
tan θ = −
8
sinθ = −
Try These:
17
15
17
secθ =
8
8
cot θ = −
15
cscθ = −
-15
a. The terminal side of an angle θ in standard position contains the point
(- 3, 4 ). Evaluate all 6 trigonometric functions for θ .
b. If csc θ = 2 and θ lies in Quadrant 2, then evaluate the five remaining
trigonometric functions.