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3.4 Circular Functions
The key trigonometric functions, sine and cosine, may be defined using the unit circle, which is symmetric
with respect to the x-axis, the y-axis, and the origin. The equation of the unit circle is x2 + y2 = 1.
y
x2 + y2 = 1
x
(1,0)
The terminal side of an angle in standard position intersects the unit circle at a unique point P(x, y).
The x-value of this point is defined to be cosine 
The y-value of this point is defined to be sine 
Because cos  and sin  are defined using the unit circle, they are called circular functions.
-1 < cos  < 1 and
-1 < sin  < 1
Example 1: Use the definitions of the sine and cosine functions to find each value.
y
x
𝜋
𝜋
(a) cos 2
(b) sin 2
𝜋
(c) cos (− 2 )
𝜋
(d) sin (− 2 )
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For any angle , the values of cos  and sin  can be found if a point Q(x, y) on the terminal side is known,
even if Q is not on the unit circle. This can be done using a reference triangle, which is found by drawing
a perpendicular from Q to the x-axis.
y
Q(x, y)
r
y
x
x
Let  be an angle in standard position, and let (x, y) be a point distinct from the origin on the
terminal side of , then:
cos 𝜃 =
𝑥
𝑟
and sin 𝜃 =
𝑦
𝑟
where 𝑟 = √𝑥 2 + 𝑦 2
Example 2: Draw the reference triangle and find the exact values of cos  and sin  of the terminal side of
an angle  in standard position passes through each point.
(a) P(2, 5)
(b) Q(3, 7)
1
(c) R(2, 2)
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Example 3: Find the exact values of cos  and sin  for angle  in standard position with the given point
on its terminal side. Draw the reference triangle.
(a) P(-3, 4)
1
(b) Q(-2, -2)
(c) R(-6, 9)
2
(d) S(-1, -5)
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Since r, which is sometimes called the radius vector is always positive, the signs of cos  and sin  are
determined by the signs of x and y. Therefore these signs depend on the quadrant in which the terminal
side lies.
y
II
I
(+,+)
(-,+)
x
(-,-)
(+,-)
III
I
Quadrant
IV
II
III
IV
cos 
+
-
-
+
sin 
+
+
-
-
Example 4: State whether each value is positive, negative, or zero.
(a) cos 45
(b) sin 3
(c) sin
5𝜋
4
(d) cos 75
(e) sin (-75)
(f) cos 5
(g) sin (-3)
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Example 5:
(a) Angle  is in standard position with its terminal side in the third quadrant. Find the exact value of cos 
1
if sin  is -2.
(b) Angle  is in standard position with its terminal side in the second quadrant. Find the exact value of
8
cos  if sin  is 10.
(c) Angle  is in standard position with its terminal side in the fourth quadrant. Find the exact value of
4
sin  if cos  is7.
Homework: Day 1: p. 144 => Class Exercises 1 – 9;
Day 2: p. 145 => Practice Exercises 1 – 22; 25 - 32
Advanced Mathematics/Trigonometry: 3.4 Circular Functions
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