Using the Unit Circle

Unit Circle

Trig Review

Multiple Choice

Set up the problem so that you can solve for X.

• sin X = 41/28

• tan X = 41/28

• cos X = 28/41

• sin X = 28/41

Multiple Choice

Solve for the missing angle

• 50.5 degrees

• 12.7 degrees

• 36.9 degrees

• 72.2 degrees

Multiple Choice

Find the measure of the indicated angle (?) round to the nearest tenth.

• 9.4
• 55.2
• 20.15
• 19.4

Multiple Choice

Susan is flying a kite, which gets caught in the top of a tree.  Use the diagram to estimate the height of the tree. 

• 63 ft
• 65 ft
• 74 ft
• 87 ft

What is the unit circle?

• It is a circle centered at the origin (0,0) with a radius of 1

• It has the formula x² + y² = 1

• It is a shortcut to finding the sine and cosine of the angles of special right triangles (30-60-90) or (45-45-90)

Why does the unit circle work?

• The distance from the center to a point on the circle is the radius, or 1.

• If we drop a perpendicular line from the point to the radius, that vertical distance is y

• If we draw a line from the center to where the perpendicular line touches the circle, that horizontal distance is x

Why does the unit circle work?

Why does the unit circle work?

How do you divide the unit circle?

Division 1

Division 2

Division 3

Reference Angle

• Measure of Terminal side to closest x axis

• Distance to x axis

• MUST be < 90 degree

• MUST be Positive

Finding Reference Angles

• The blue portion of graph is the original angle

• The red portion is reference angle

• 
  

• 
  

Finding Reference angle by Quadrant

• Quadrant 1 - itself

• Quadrant 2 - 180 Degrees minus the angle

• Quadrant 3 - The angle minus 180 degrees

• Quadrant 4 - 360 minus the angle

Multiple Choice

What is the reference angle?

• 112

• 68

• 158

• 248

Multiple Choice

What is the reference angle?

• 43

• -43

• 47

• -47

Multiple Choice

What is the reference angle for 63°?

• 117°

• 207°

• 63°

• 153°

Multiple Choice

What is the reference angle for -30°?

• 150°

• 30°

• 80°

• 60°

Multiple Choice

What is the reference angle for 125°?

• 235°

• 125°

• 75°

• 55°

What are the values for sine and cosine for each angle on the unit circle?

Values for Cosine and Sine

• The cosine and sine of each angle are the (x, y) values.

• Cosine is the x value

• Sine is the y-value

• The signs correspond to the sign of the quadrants of the coordinate plane

Values for the Cosine and Sine

A Couple of Things to Notice

Multiple Choice

At what point are sine and cosine the same value?

• 0

• 90

• 45

• 60

Multiple Choice

What is the value of sine at 330

• 1/2

• -1/2

• √3 / 2

• -√3 / 2

Multiple Choice

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

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

• (−√2∕2, √2∕2)

• (−√2∕2, −√2∕2)

• (−√3∕2, −√2∕2)

• (√2∕2, −√2∕2)

Multiple Choice

sin π/4

(Multiply by   $$\frac{180}{\pi}$$  )

• √2∕2

• 1/2

• √3/2

• 0

Multiple Choice

cosθ is positive in

• the 1st and 2nd quadrants
• the 1st and 3rd quadrants
• the 1st and 4th quadrants
• the 2nd and 3rd quadrants

Multiple Choice

What is the radius of a unit circle?

• 1
• 2
• 1/2
• 0

Multiple Choice

sinθ is positive in

• the 1st and 2nd quadrants
• the 1st and 3rd quadrants
• the 1st and 4th quadrants
• the 2nd and 3rd quadrants

Multiple Choice

sin 7π/6

(Multiply by  $$\frac{180}{\pi}$$  )

• 1/2

• -1/2

• √3/2

• -√3/2

Multiple Choice

cos(60°)

• 0

• 1/2

• √3/2

• 1

Multiple Choice

cos(120°)

• 0

• 1/2

• √3/2

• 1

Multiple Choice

Which degree is equivalent to $$\frac{\pi}{4}$$  ?

• $$210\degree$$  

• $$45\degree$$  

• $$150\degree$$  

• $$60\degree$$  

Multiple Choice

How many radians make up a SEMICIRCLE?

• 90

• $$\pi$$

• $$2\pi$$

• $$\frac{3\pi}{4}$$

Multiple Choice

What is the coordinate point at $$225\degree$$  on the unit circle?

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

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

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

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

Multiple Choice

What is the radian measure at  $$315\degree$$  ?

• $$\frac{3\pi}{4}$$  

• $$\frac{\pi}{2}$$  

• $$\frac{7\pi}{4}$$  

• $$\frac{7\pi}{6}$$  

Multiple Choice

True or False. Only the cosine trig functions is positive in Quadrant II.

• True

• False

Multiple Choice

What is the order of the coordinate pairs for the points on the Unit Circle?

• $$\left(\tan\theta,\sin\theta\right)$$

• $$\left(\cot\theta,\cos\theta\right)$$

• $$\left(\sin\theta,\cos\theta\right)$$

• $$\left(\cos\theta,\sin\theta\right)$$

Fill in the Blanks

The degree equivalent to $\frac{3\pi}{2}$  is

$\degree$

(DO NOT TYPE THE DEGREE SYMBOL)

Multiple Choice

What is $$\cos\left(\frac{5\pi}{6}\right)$$  ?

• $$-\frac{1}{2}$$

• $$\frac{1}{2}$$

• $$\frac{\sqrt{3}}{2}$$

• $$-\frac{\sqrt{3}}{2}$$

Multiple Choice

What is the coordinate point at  $$\frac{4\pi}{3}$$  ?

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

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

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

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

Multiple Choice

True or False. All trig functions are positive in Quadrant IV.

• True

• False