Skill 9
The Unit Circle
Skill 9a: Find the value of trig functions given a point on the terminal side of the angle
Skill 9b: Given an angle find coterminal Angles and Reference Angles
Skill 9c: Given one trig ratio and a quadrant, find the exact values of other trig ratios
Skill 9d: Use coterminal and reference angles to find exact values of trig function
Skill 9e: Use the unit circle to find value of trig functions at special angles
Skill 9a: Find the value of trig functions given a point on the terminal side of the angle
Plot the ordered pair (6, 8). Draw a right triangle in standard postion with this point on the terminal side.
Find the following:
A) sin(𝜃) = ________
B) cos(𝜃) = ________
C) tan(𝜃) = ________
D) sec(𝜃) = ________
E) csc(𝜃) = ________
F) cot(𝜃) = ________
The values for the six trigonometric functions can be positive or negative, depending on location of the ordered
pair. The values are determined using a right triangle drawn to the x-axis (either positive or negative).
1. Find the values for the six trig functions for the angle in standard position whose terminal side passes through
the point (2, -6).
A) sin(𝜃) = ________
B) cos(𝜃) = ________
C) tan(𝜃) = ________
D) sec(𝜃) = ________
E) csc(𝜃) = ________
F) cot(𝜃) = ________
2. Find the values for the six trig functions for the right triangle with one leg on the x-axis and the hypotenuse
passing through the point (-7, -7).
A) sin(𝜃) = ________
B) cos(𝜃) = ________
C) tan(𝜃) = ________
D) sec(𝜃) = ________
E) csc(𝜃) = ________
F) cot(𝜃) = ________
9b: Given an angle find Coterminal Angles and Reference Angles
Recall that two angles in standard position are coterminal if their sides coincide. For example, an angle of 60˚, 300˚, and 420˚ are all coterminal.
Find an angle with a measure between 0˚ and 360˚ or 0 and 2π that is coterminal with the given angle.
1. 710˚
2. -833˚
3.
17𝜋
5
4. −
8𝜋
3
A reference angle is the acute angle formed between the terminal side of the angle and the x-axis.
To find a reference angle, first determine the quadrant location of the angle. Find a coterminal angle with a
measure between 0˚ and 360˚ or 0 and 2π to determine this.
Quadrant 2
Quadrant 1
Reference Angle = 180˚ (or π) - Coterminal Angle
Reference Angle = Coterminal Angle
Quadrant 3
Quadrant 4
Reference Angle = Coterminal Angle - 180˚ (or π)
Reference Angle = 360˚ (or 2π) - Coterminal Angle
Determine the quadrant of each angle. Then determine the reference angle.
5. 167˚
9.
17𝜋
10
6. 315˚
10.
10𝜋
3
7. - 129˚
8. 490˚
11.
12. 10.337
-1.412
9c: Given one trig ratio and a quadrant, find the exact values of other trig ratios
4
In the diagram at the right, sin(𝜃) = 5. Use this information to
determine possible values for BC, AC and AB.
Based on these values, determine the following:
A) cos(𝜃) = ________
B) tan(𝜃) = ________
C) sec(𝜃) = ________
D) csc(𝜃) = ________
E) cot(𝜃) = ________
12
1. The cosine of an angle located in Quadrant 2 is − 13. Sketch the angle and determine the following:
A) sin(𝜃) = ________
B) tan(𝜃) = ________
C) sec(𝜃) = ________
D) csc(𝜃) = ________
2. The tangent of an angle located in Quadrant 3 is
A) sin(𝜃) = ________
B) tan(𝜃) = ________
C) sec(𝜃) = ________
D) csc(𝜃) = ________
4. sec(𝜃) =
√5
2
E) cot(𝜃) = ________
√5
.
2
Sketch the angle and determine the following:
E) cot(𝜃) = ________
If 𝜃 is located in Quadrant 4, Sketch the angle and determine the following:
A) sin(𝜃) = ________
B) tan(𝜃) = ________
C) sec(𝜃) = ________
D) csc(𝜃) = ________
E) cot(𝜃) = ________
9d: Use coterminal and reference angles to find exact values of trig functions
Since the value for sine depends on the vertical side of the right triangle and the value of cosine depends on the
horizontal side of the right triangle, the signs of the values for sine (and cosecant), cosine (and secant), and
tangent (and cotangent) vary based on the quadrant the angle is located in.
sine +
sine +
cosine -
cosine +
tangent -
tangent +
sine -
sine -
cosine -
cosine +
tangent +
tangent -
T
he value of the six trig functions at any angle is the same as the value of the trig functions for the reference angle,
adjusted with the correct sign conventions above.
Determine the reference angle for each angle and determine the following for the angle and the reference
angle.
𝜋
6
1. 210˚
2. 135˚
3. −
sin(210˚) = ______
sin(135˚) = ______
sin(− 6 ) = ______
sin(8.113) = ______
sin(_____) = ______
sin(____) = ______
sin(_____) = _____
sin(____) = _______
4. 8.113
𝜋
Given that sin(18) = 0.3090 and cos(18) = 0.9511, determine the following. Verify the results using a calculator
5. 558˚
6. -18˚
7. 162˚
A) sin(𝜃) = ________
A) sin(𝜃) = ________
A) sin(𝜃) = ________
B) cos(𝜃) = ________
B) cos(𝜃) = ________
B) cos(𝜃) = ________
Determine the reference angle for each angle and determine the exact value of the following for each angle.
3𝜋
7𝜋
5𝜋
)
6
8. cos⁡( 4 )
9. sin⁡( 3 )
10. csc⁡(−
A) sin(𝜃) = ________
A) sin(𝜃) = ________
A) sin(𝜃) = ________
B) cos(𝜃) = ________
B) cos(𝜃) = ________
B) cos(𝜃) = ________
C) tan(𝜃) = ________
C) tan(𝜃) = ________
C) tan(𝜃) = ________
Values of Sine, Cosine, and Tangent on the x and y-axis.
Draw a right triangle in standard position with 𝜃 = 0
Determine the following:
A) sin(0) = ________
B) cos(0) = ________
C) tan(0) = ________
Determine the following:
𝜋
𝜋
11A) sin(𝜋2) = ________
11B) cos ( 2 ) = ________
11C) tan ( 2 ) = ________
12A) sin(𝜋) = ________
12B) cos(𝜋) = ________
12C) tan(𝜋) = ________
13A) sin(3𝜋
) = ________
2
13B) cos ( 2 ) = ________
3𝜋
3𝜋
13C) tan ( 2 ) = ________
9e: Use the unit circle to find value of trig functions at special angles
The unit circle is the set of all points located a distance of 1 away from the origin.
1. Verify that each point shown is located on the unit circle.
A)
A
B
B)
C)
C
.
2
2. The x-coordinate of a point on the unit circle is 3. Determine the value of the y-coordinate for the
point if it is located in Quadrant 4.
Draw the right triangle that is formed using the point on shown on the unit circle as a point on the
hypotenuse. Label the angle that is in standard position 𝜃.
1
3. Show that cos(𝜃) = 2
4. Show that sin(𝜃) =
√3
2
For any right triangle drawn using the a point on the unit circle as a point on the hypotenuse, the angle
formed with the x-axis has a value of cosine that is the x-coordinate of the point and the value of sine is
the y-coordinate of the point.
P is a point on the unit circle. Determine the coordinates of P for the following values of 𝜽
𝜋
5. 𝜃 =
6
𝜋
7. 𝜃 =
2
9. 𝜃 = 330°
𝜋
6. 𝜃 = ⁡ 4
8. 𝜃 = ⁡225°
10. 𝜃 = ⁡
11𝜋
4
The values of sine and cosine MUST BE MEMORIZED for the following angles and all angles
corresponding angles. Notice the pattern that is formed by these values in the table below. This can
help in memorizing.
sin⁡(𝜃)
𝜃
0
cos⁡(𝜃)
𝜋
𝜋
𝜋
30°,
6
𝜋
45°,
4
𝜋
60°,
3
𝜋
90°,
2
90°, 2
60°, 3
𝜋
45°, 4
𝜋
30°, 6
0
Your hand can also be a helpful way to remember these values.
1) Hold down the finger of the angle you want to know
2) Number of fingers above is the numerator of cosine
3) Number of fingers below is numerator of sine