4.2 Trigonometric Functions: The Unit Circle

​4.2 Trigonometric Functions: The Unit Circle

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Using the Unit Circle

• To use the Unit Circle to evaluate cosine, sine, or tangent we use the coordinates of the point of intersection between the terminal side of the angle and the Unit Circle.

• The x-coordinate of the point is equal to the cosine of the angle.

• The y-coordinate of the point is equal to the sine of the angle.

• To find the tangent of an angle you take the y-coordinate/x-coordinate.

Evaluate Cosine

• Cos(angle) = x-coordinate of point

• Cosine is positive in the FIRST and FOURTH quadrants.

• Cosine is negative in the SECOND and THIRD quadrants.

• cos(2π/3) = -1/2

• cos(270°) = 0

• cos(π/6) = √3/2

Evaluate Sine

• Sin(angle) = y-coordinate of point

• Sine is positive in the FIRST and SECOND quadrants.

• Sine is negative in the THIRD and FOURTH quadrants.

• sin(135°) = √2/2

• sin(4π/3) = -√3/2

• sin(0) = 0

Multiple Choice

What is cos 60°? #1

• ½

• ^√3/₂

• ^√2/₂

• 1

Multiple Choice

What is sin 45°? #2

• 1

• ^√3/₂

• ½

• ^√2/₂

Evaluate Tangent

• Tan(angle) = y-coordinate/x-coordinate

• Tangent is positive in the FIRST and THIRD quadrants.

• Tangent is negative in the SECOND and FOURTH quadrants.

• tan(90°) = 1/0 = UNDEFINED

• tan(π) = 0/-1 = 0

• tan(240°) = (-√3/2)/(-1/2)

• =(-√3/2) *(-2/1) = √3/1 = √3

Multiple Choice

What is tan 45°? #5

• 1

• ^√2/₂

• ^√3/₂

• ½

​Here is an overview of all major angles in radians on the unit circle:

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Multiple Choice

Evaluate $$\tan\left(\frac{7\pi}{6}\right)$$   . #14

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$  

Multiple Choice

Evaluate $$\sin\left(\frac{23\pi}{6}\right)$$   . #15

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$  

Multiple Choice

Find the equivalent value for $$\cot\ -300\degree.$$  #19

• $$-\cot60\degree$$  

• $$\frac{1}{\tan\ 60\degree}$$  

• $$\frac{1}{\tan300\degree}$$  

• $$\cot120\degree$$  

Multiple Select

Which of the following is a point on the unit circle? #20

(Check all that apply)

• $$\left(1,\ 1\right)$$  

• $$\left(-1,\ 0\right)$$  

• $$\left(\frac{1}{\sqrt{2}},\ \frac{1}{\sqrt{2}}\right)$$  

• $$\left(\frac{1}{\sqrt{5}},-\frac{2}{\sqrt{5}}\right)$$  

Multiple Choice

What is the correct ordered pair for an angle rotation of  $$150\degree$$  ? #21

• $$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$$  

• $$\left(\frac{-\sqrt{3}}{2},\ \frac{1}{2}\right)$$  

• $$\left(\frac{-\sqrt{3}}{2},\frac{-1}{2}\right)$$  

• $$\left(\frac{\sqrt{3}}{2},\frac{-1}{2}\right)$$  

Multiple Choice

What are the exact coordinates of 3π/4 on the unit circle?

(Multiply by  $$\frac{180}{\pi}$$  to get degrees) #37

• (−√2∕2, √2∕2)

• (−√2∕2, −√2∕2)

• (−√3∕2, −√2∕2)

• (√2∕2, −√2∕2)

Match

Match the following angles measured in radians to degrees. #41

• $$90\degree$$  

•    $$180\degree$$

• $$30\degree$$

• $$45\degree$$

• $$60\degree$$

 

• $$\frac{\pi}{2}$$  

• $$\pi$$

• $$\frac{\pi}{6}$$

• $$\frac{\pi}{4}$$

• $$\frac{\pi}{3}$$

Match

Match the correct value of the trigonometric ratios of special angles. #42

• $$\frac{\sqrt[]{2}}{2}$$  

•    $$\frac{\sqrt[]{3}}{2}$$

• $$\frac{\sqrt[]{3}}{3}$$

• $$\frac{1}{2}$$

• $$\frac{1}{2}$$

 

• $$\sin\frac{\pi}{4}$$  

• $$\cos\frac{\pi}{6}$$

• $$\tan\frac{\pi}{6}$$

• $$\cos\frac{\pi}{3}$$

• $$\sin\frac{\pi}{6}$$

Multiple Choice

What is $$\sec\left(\frac{7\pi}{4}\right)$$  ? #7

• $$\frac{\sqrt{2}}{2}$$  

• $$\sqrt{2}$$  

• $$-\frac{\sqrt{2}}{2}$$  

• $$-\sqrt{2}$$  

Multiple Choice

Find the sin(-9π/4). #8

• -√3/2

• -1/2

• √2/2

• -√2/2

Multiple Choice

What is the tan(7π/6)? #9

• _√3

• -√3/3

• √3/3

• 1

Multiple Choice

What is the csc (7π/6)? #10

• -2

• 2

• -1/2

• 1/2