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Math 1330
Section 4.3
Section 4.3: Unit Circle Trigonometry
An angle is formed by two rays that have a common endpoint (vertex). One ray is called the initial
side and the other the terminal side. A terminal angle can lie in any quadrant, on the x-axis or yaxis.
An angle is in standard position if the vertex is at the origin of the two-dimensional plane and its
initial side lies along the positive x-axis.
Positive angles are generated by counterclockwise rotation. Negative angles are generated by
clockwise rotation.
An angle in standard position whose terminal side lies on either the x-axis or the y-axis is called a
quadrantal angle.
Math 1330
Section 4.3
Example 1: Draw each angle in standard position.
a. 240°
c.
b. −150°
7π
3
d.
− 5π
4
Angles in standard position with the same terminal side are called coterminal angles. The measures
of two coterminal angles differ by a factor corresponding to an integer number of complete
revolutions.
The degree measure of coterminal angles differ by an integer multiple of 360o . For any angle θ
measured in degrees, an angle coterminal with θ can be found by the formula θ + n *360o .
The radian measure of coterminal angles differ by an integer multiple of 2π . For any angle θ
measured in radians, an angle coterminal with θ can be found by the formula θ + n * 2π .
Example 2: Find three angles, two positive and one negative that are co-terminal with each angle.
a. θ = 45o .
b.
θ =−
π
6
Math 1330
Section 4.3
The Reference Angle or Reference Number
Let θ be an angle in standard position. The reference angle associated with θ is the acute angle
(with positive measure) formed by the x-axis and the terminal side of the angle θ . When radian
measure is used, the reference angle is sometimes referred to as the reference number (because a
radian angle measure is a real number).
Example 3: Find the reference angles for each of these angles.
5π
a. Sketch the angle
and find its reference angle.
6
b. Sketch the angle
4π
and find its reference number.
3
Math 1330
Section 4.3
c. Sketch the angle 300o and find its reference angle.
d. Sketch the angle −210o and find its reference angle.
To define the trigonometric functions, we begin by placing the angle θ in standard position and
drawing in the unit circle x 2 + y 2 = 1 . The point P(x, y) is the point where the angle intersects the
unit circle.
Trigonometric Functions of Angles
Y
x + y =1
2
P(x, y)
θ
sin θ = y
tan θ =
Note: (x, y) = (cos θ , sin θ )
Note:
y
OR
(+, +)
(+, -)
( x ≠ 0)
( x ≠ 0)
( y ≠ 0)
( y ≠ 0)
All Students Take Calculus
y
S
x
(-, -)
y
x
1
x
1
csc θ =
y
x
cot θ =
y
sec θ =
2
x
(-, +)
cos θ = x
A
OR
x
T
C
This should help you to know which trigonometric functions are positive in which quadrant.
Math 1330
Section 4.3
The six trig functions of θ are defined as follows, using the circle on the previous page.
x
r
y
sinθ =
r
y
tanθ = ( x ≠ 0)
x
cosθ =
r
x
r
cscθ =
y
x
cot θ =
y
secθ =
( x ≠ 0)
y = opposite side
x = adjacent side
y = hypotenuse
( y ≠ 0)
( y ≠ 0)
P(x,y)
r
y
x
Note: If θ is a first quadrant angle, these definitions are consistent with the definitions given in
Section 4.1
We will most often work with a unit circle with radius 1. In this case, each value of r is 1. This
adjusts the trig functions as follows:
Note: (x, y) = (cos θ , sin θ )
An identity is a statement that is true for all values of the variable. Here are some identities that
follow from the definitions above.
tanθ =
sinθ
cosθ
tanθ =
1
cot θ
cot θ =
cosθ
sinθ
cot θ =
1
tanθ
cscθ =
1
sinθ
secθ =
1
cosθ
Math 1330
Section 4.3
Note:
π
2
y
(-, +)
(+, +)
x
(-, -)
or 90°
π or 180°
(cosθ , sin θ )
0 or 360°
(+, -)
3π
or 270°
2
Example 4: Name the quadrant in which both conditions are true.
a. csc < 0 and tan > 0
b. cos < 0 and csc > 0
Example 5: Evaluate each of the following trig functions using the reference angles.
a. tan 210°
b. sin (140°)
c. sec (−
ଶగ
ଷ
)
d. sin(330 o ) + sec(135 o )
.
Math 1330
Section 4.3
Trig Functions of Quadrant Angles and Special Angles
You will need to find the trig functions of quadrant angles and of angles measuring 30°,45°or 60°
without using a calculator.
Here is a simple way to get the first quadrant of trigonometric functions. Under each angle measure,
write down the numbers 0 to 4. Next take the square root of the values and simplify if possible.
Divide each value by 2. This gives you the sine values of each of the angles you need. To fine the
cosine values, write the previous line in reverse order. Now find the tangent values by using the sine
and cosine values
Angle in
Degrees
Angle in
Radians
0
30
45
60
90
0
ߨ
6
ߨ
4
ߨ
3
ߨ
2
Sine
Cosine
Tangent
Example 5: Let the point P(x, y) denote the point where the terminal side of angle θ (in standard
position) meets the unit circle. Evaluate the six trig functions.
π
4
Suppose that x = − and < θ < π . Find all 6 trig functions.
5
2
Math 1330
Section 4.3
Unit Circle
Evaluating Trigonometric Functions Using Reference Angles
1. Determine the reference angle associated with the given angle.
2. Evaluate the given trigonometric function of the reference angle.
3. Affix the appropriate sign determined by the quadrant of the terminal side of the angle
in standard position.
Example 6: Evaluate the following.
a. sin (300°)
b. cos (−240°)
Math 1330
Section 4.3
c. sec(135°)
3ߨ
d. cos ൬ ൰
4
e. sin ൬
4ߨ
൰
3
f. tan(90°)
g. cot(−7ߨ)
Math 1330
Section 4.3
11ߨ
h. sin ൬−
൰
2
5ߨ
tan ቀ 6 ቁ − 2 cos(2ߨ)
i.
3ߨ
4 sin ቀ− ቁ
2