Trig & the Unit Circle Exam Review

Match

Match the following

• Circle with a radius of 1, centered on the origin (0,0)

• A positive acute angle formed by its terminal side and the x-axis

• An angle whose terminal side lies on the x-axis or y-axis

• Formula used to relate the sides of a right triangle

• Angles in standard position that share the same terminal side

Unit Circle

Reference Angle

Quadrantal Angles

Pythagorean Theorem

Coterminal Angle

• Unit Circle

• Reference Angle

• Quadrantal Angles

• Pythagorean Theorem

• Coterminal Angle

Vocabulary (Mix & Match)
Trig Values from a right triangle (2)
Trig values from a point on a coordinate plane (2)
Signs of trig values (2)
Trig values from a trig value & quadrant (2)
Reference angles (2)
Unit Circle (5)

Unit circle
reference angle
quadrantal anlges
pythagoean thorem
coterminal angle

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Signs of Trig Values

I II III IV

All Students Take Calculus

​All are +.

​Sine is +.

Tangent is +.

Cosine is +.

Multiple Select

Check the quadrants in which the y-value of the ordered pair on the unit circle is negative.

• Q1

• Q2

• Q3

• Q4

Multiple Choice

What is the sign of the x-coordinate in Quadrant 3?

• Positive

• Negative

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



sine is positive: ​

Placeholder

​&

Placeholder

sine is negative: ​ ​

Placeholder

&​

Placeholder

Drag these tiles and drop them in the correct blank above

I

II

III

IV

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



cosine is positive: ​

Placeholder

​&

Placeholder

cosine is negative: ​ ​

Placeholder

&​

Placeholder

Drag these tiles and drop them in the correct blank above

I

IV

II

III

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



tangent is positive: ​

Placeholder

​&

Placeholder

tangent is negative: ​ ​

Placeholder

&​

Placeholder

​

Drag these tiles and drop them in the correct blank above

I

III

II

IV

• ​A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. 

• A reference angle is always positive and is always less than 90º.

• To calculate - we measure THE DISTANCE TO/FROM THE X-AXIS!

• This helps us when we have angles larger that 360 degrees. We can simplify the work if we can refer to a smaller angle.

​Reference Angles

Multiple Select

Which is true about reference angles?

• They're acute

• Formed using the y-axis and terminal side.

• Formed using the x-axis and terminal side.

• Used to calculate with large angles.

• They're right angles.

1. Determine the quadrant location of the given angle

   1. You will have to find a co-terminal angle for angles above 360⁰ or negative angles.

2. Use the formula for the determined quadrant.

   1. For Example: If a given angle is in Quadrant II --> θ_r=180-θ

​How to Find a Reference Angle

​Remember:  The reference angle is measured from the terminal side of the original angle "to" the x-axis (not "to" the y-axis).

Multiple Choice

What is the reference angle for 63°?

• 117°

• 207°

• 63°

• 153°

Multiple Choice

What is the reference angle?

• 112

• 68

• 158

• 248

Multiple Choice

What is the reference angle for 125°?

• 235°

• 125°

• 75°

• 55°

Multiple Choice

What is the reference angle?

• 43

• -43

• 47

• -47

Multiple Choice

What is the reference angle?

• 163

• 73

• 27

• 107

Multiple Choice

What is the reference angle for 275°?

• 85°

• 95°

• -85°

• -95°

Dropdown

Look at YOUR unit circle. What is the reference angle for $$\frac{4\pi}{3}$$ ?



Reference angle = ​

Select answer

Dropdown

Look at YOUR unit circle. What is the reference angle for $$\frac{7\pi}{4}$$ ?



Reference angle = ​ ​

Select answer

Multiple Select

Which of the following have the same reference angle? Select all that apply.

• $$105\degree$$  

• $$435\degree$$  

• $$205\degree$$  

• $$285\degree$$  

Multiple Choice

What is the reference angle for -30°?

• 150°

• 30°

• 80°

• 60°

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 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

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 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

Dropdown

Find the exact value of the following trig expressions.



Try to remember these results without looking at the unit circle.



$$\sin\left(\frac{\pi}{3}\right)=$$ ​

Select answer

$$\sin\left(\frac{7\pi}{6}\right)=$$ ​ ​ ​

Select answer

$$\cos\left(\frac{5\pi}{4}\right)=$$ ​

Select answer

$$\cos\left(-\frac{\pi}{3}\right)=$$ ​

Select answer

Multiple Choice

What trig functions make up the coordinates in the unit circle?

• cosine and sine

• only sine

• only cosine

• sine, cosine, and tangent

Match

Match the following. Let θ be an angle such that $$\sec\theta=-\frac{5}{4}$$ and $$\tan\theta>0$$ . Find the exact values of cos θ, cot θ and csc θ

• $$\cot\theta$$

• $$\csc\theta$$

• $$\cos\theta$$

• $$\frac{4}{3}$$

• $$-\frac{5}{3}$$

• $$-\frac{4}{5}$$

Match

Match the following. Let θ be an angle in quadrant IV such that $$\tan\theta=-\frac{5}{12}$$ . Find the exact values of sin θ, cos θ and csc θ

• $$\cos\theta$$

• $$\csc\theta$$

• $$\sin\theta$$

• $$\frac{12}{13}$$

• $$-\frac{13}{5}$$

• $$-\frac{5}{13}$$

Match

Match the following. Let θ be an angle in quadrant III such that $$\sin\theta=-\frac{1}{4}$$

• $$\sec\theta$$

• $$\tan\theta$$

• $$\cot\theta$$

• $$\frac{-4\sqrt[]{15}}{15}$$

• $$\frac{\sqrt[]{15}}{15}$$

• $$\sqrt[]{15}$$

Match

Match the following.

Let (-5, 2) be a point on the terminal side of θ. Find the exact values of cos θ, csc θ, and tan θ.

• $$\cos\theta$$

• $$\csc\theta$$

• $$\tan\theta$$

• $$\frac{-5\sqrt[]{29}}{29}$$

• $$\frac{\sqrt[]{29}}{2}$$

• $$-\frac{2}{5}$$

Multiple Choice

Evaluate cos(135°)

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

Multiple Choice

Evaluate cos(11π/6)

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

Fill in the Blank

Evaluate cos(90°)

Multiple Choice

Evaluate sin(60°)

• 0

• $$\frac{1}{2}$$  
  

• $$\frac{\sqrt{2}}{2}$$  
  

• $$\frac{\sqrt{3}}{2}$$  

• 1

Multiple Choice

sin 180°

• 1

• -1

• 0

• $$-\frac{\sqrt{3}}{2}$$

• $$\frac{1}{2}$$

Multiple Choice

Evaluate cos(225°)

• $$-\frac{\sqrt{2}}{2}$$

• $$\frac{\sqrt{2}}{2}$$

• $$-\frac{\sqrt{3}}{2}$$

• $$-\frac{1}{2}$$

Multiple Choice

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

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

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

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

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

Fill in the Blank

The unit circle is called the unit circle becasue it has a radius of ____ units.

Multiple Choice

True/False: The Unit Circle center is at the origin.

• True

• False