Section 3 – Circular Functions
Objective
To find the values of the six trigonometric
functions of an angle in standard position
given a point on the terminal side.
Sine and Cosine Functions
Using the Unit Circle
Consider an angle q in standard position.
The terminal side of the angle intersects
the circle at a unique point, P(x, y). The
y-coordinate of this point is sine q. The
x-coordinate is cosine q.
y
P(x, y)=
P(cosx, sinx)
The abbreviation for sine is sin.
The abbreviation for cosine is cos.
x
q
Definition of Sin and Cos
If the terminal side of an angle q in standard position
intersects the unit circle at P(x, y), then cos q = x and
(0, 1)
sin q = y.
Find sin 90°
90°
Remember, the unit circle
has a radius of 1.
The terminal side of a 90° angle in
standard position is the positive y-axis,
what are the coordinates of the point
where it intersects the unit circle?
Since the sin q = y, what does
sin 90° = 1
Sin and Cos
Find cos p
The terminal side of an
angle of p radians is the (-1, 0)
negative x-axis. What
are the coordinates of
the point?
p
Since sin q = y. The cos p = 0
Sin and Cos
When we use the unit circle, the radius, r, is 1.
Since sin q = y, we can say sin q =
Since cos q = x, we can say cos q =
y
r
x
r
P(x, y)
1
Sin and Cos Functions of Any
Angle in Standard Position

For any in standard position with
measure q, a point P(x, y) on its
terminal side, and r = √ x² + y², the sin
and cos functions of q are as follows:
sin q = y
r
cos q = x
r
Find the values of the sin and cos functions of
an angle in standard position with measure q if
the point with coordinates (3, 4) lies on its
terminal side.
You know that x = 3 and y = 4.
You need to find r.
r = √ 3² + 4²
r = √9 + 16
=5
(3, 4)
r
4
3
Now you know x = 3, y = 4, and r = 5.
You can write the sin and cos functions.
Sin = y/r, or 4/5.
Cos = x/r, or 3/5.
Find sin q when cos q = 5/13 and the
terminal side of q is in Quadrant 1.
Since cos q = x/r = 5/13 and r is always positive,
r = 13 and x = 5.
You now have to find y.
13
r = √x² + y²
13 = √5² + y²
169 = 25 + y²
5
144 = y²
+ 12 = y
Since y is in Quadrant 1,
y must be positive.
So, sin q = y/r, or 12/13
y
There are 4 other trig functions – tangent (tan),
cotangent (cot), secant (sec),
and cosecant (csc).

For any angle in standard position with measure q,
and a point P(x, y) on its terminal side the trig
functions are:
sin q = y/r
cos q = x/r
tan q = y/x
csc q = r/y
sec q = r/x
cot q = x/y
The terminal side of an angle in standard
position contains the point with coordinates
(8, -15). Find tan, cot, sec, and csc.

Since x = 8 and y = -15, you must find r.
r = √8² + (-15)² x = 8, y = -15, and r = 17.
= √289
You can now write the trig functions.
= 17
tan q = -15/8
8
-15
r
cot q = -8/15
sec q = 17/8
csc q = -17/15
If csc q = -2 and q lies in Quadrant 3, find
sin, cos, tan, cot, and sec.
Since csc and sin are reciprocals, sin q = -½.
To find the other functions, you have to find the coordinates of a
point on the terminal side.
Since sin q = -½ and r is always positive, let r = 2 and y = -1.
Find x.
Since the terminal side of
r² = x² + y²
q lies in quadrant 3,
2² = x² + (-1)²
x = -√3
4 = x² + 1
sin q = -√3/2
3 = x²
√3 = x
tan q = -1/-√3
sec q = 2/-√3 or 2√3/3
cot q = -√3/ -1
Assignment

Page 260 – 261

#22 – 41, 49, 50