Trigonometry and Unit Circle Review

8.6 Trig Review

​

Unit Circle

We build the unit circle by creating triangles with a radius of 1.

The coordinates (cos, sin) of each point on the unit circle that aligned with the angle

You will need to be able to:

find the x (cos) and y (sin) value for various angles on the unit circle

Multiple Choice

At what positive angle are sine and cosine the same value? 

• $$45\degree$$  

• $$60\degree$$  

• $$90\degree$$  

• $$135\degree$$  

Multiple Choice

What is the value of cosine at 135 degrees

• 0

• $$\frac{1}{2}$$  

• $$\frac{\sqrt[]{2}}{2}$$  

• $$\frac{-\sqrt[]{2}}{2}$$  

Multiple Choice

what is the exact value of cos for this angle?

$$\frac{2\pi}{3}\ or\ 120\degree$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{-\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

Multiple Choice

Based on your unit circle

cos(0^o) =

• 1

• 0

• −1

• 1/2

Radians

Unit of measure for angles as an altrenative to degrees.

​

1 radian is created, when the radius and the arc length are the same.

​

there is 2 pi radians in a circle

there is 1 pi radians in half a circle. ​

​

​Angles in standard postion always start at the same spot on the unit circle.

​Positive angles go up, negative angles go down. ​

​

Divide the half circles into the number of pieces shown in the denominator

The angle will go the number of pieces in the numerator​

​

Multiple Choice

Select the correct sketch of the angle in standard position:  $$\frac{7\pi}{4}$$  

Multiple Choice

Select the graph of the angle in standard position: $$\frac{2\pi}{3}$$  

Multiple Choice

Select the graph of the angle in standard position:  $$\frac{5\pi}{3}$$  

Multiple Choice

Select the graph of the angle in standard position: $$\frac{3\pi}{4}$$  

Multiple Choice

Select the graph of the angle in standard position: $$\frac{5\pi}{4}$$  

Multiple Choice

Select the graph of the angle in standard position: $$\frac{4\pi}{3}$$  

Multiple Choice

Find the measure of the angle, in radians.

• $$\frac{\pi}{4}$$

• $$2\pi$$

• $$\frac{\pi}{2}$$

• $$\frac{3\pi}{4}$$

​convert radians to degrees

​

multiply by 180/pi​

Multiple Choice

To convert from radians to DEGREES

• multiply by

  $$\frac{π}{180°}$$

• multiply by $$\frac{180°}{π}$$

• multiply by $$\frac{π}{\sqrt{2}}$$

• multiply by $$\frac{π}{\sqrt{3}}$$

Multiple Choice

To convert from degrees to RADIANS

• multiply by

  $$\frac{1}{180°}$$

  and put pi on your answer

• multiply by $$\frac{180°}{π}$$

• multiply by $$\frac{π}{\sqrt{2}}$$

• multiply by $$\frac{π}{\sqrt{3}}$$

Multiple Choice

What is 240 degrees in radians? 

• π/3

• 4π/3

• 5π/3

• 2π/3

Multiple Choice

What is 5π/6 radians in degrees?

• 180

• 150

• 360

• 120

Multiple Choice

What is 225  degrees in radians? 

• 7π/4
• 5π/4
• 3π/4
• 5π/3

Multiple Choice

What is 7π/4 radians in degrees?

• 300
• 135
• 315
• 45

Multiple Choice

What is 45 degrees in radians?

• π/2
• π/6
• π/4
• 3π/4

Multiple Choice

What is 2π/3 radians in degrees? 

• 150
• 120
• 180
•  30

The sine function intersects with the midline​ at the y axis. One cycle of sin function is a hill and then a valley

​

The cos function is above the midline​ at the y axis. One cycle of the cos function is a rounded V shape

​Amplitude

First identify the midline.

The amplitude is distance from the midline to the highest or lowest point of the graph.

​When looking at a graph and trying to determine its equation.

1. find midline = d

2. cos or sin

3. find amplitude =a

4. ​find # of cycles in 2pi or 360 degrees = b or

5. find length of one cycle period​.

   2pi/period = b​

Multiple Choice

Sine or Cosine?

• sine

• cosine

Multiple Choice

Sine or Cosine?

• y = sinx

• y = cosx

• y = tanx

Multiple Choice

sine or cosine?

• cosine

• sine

Multiple Choice

Is this sine or cosine?

• Sine

• Cosine

• Tangent

Multiple Choice

Sine or Cosine?

• Cosine

• Sine

Multiple Choice

First find the midline,

then determine the amplitude.

The amplitude is:

• 2

• 4

• -2

• 1

Multiple Choice

First find the midline,

then determine the amplitude.

The amplitude is:

• 4

• 3

• 2

• 6

Multiple Choice

What is the b?

(the number cycles from 0 to 2pi on x axis)

• 3

• 2

• 1

• 4

Multiple Choice

What is the b?

(the number cycles from 0 to 2pi on x axis)

• b = 3

• b = 2

• b =1

• b = 4

Multiple Choice

What is the period?

(the length of one cycle, start at 0 on x axis)

• $$2\pi$$  

• $$4\pi$$  

• $$1\pi$$  

• $$8\pi$$  

Multiple Choice

What is the period

(the length of one cycle, start at 0 on x axis)

• $$4\pi$$  

• $$3\pi$$  

• $$2\pi$$  

• $$1\pi$$  

Multiple Choice

What is the function?

(you can always graph the answer choices in Desmos)

• f(x) = sin(1/2x)

• f(x) = sin(2x)

• f(x) = cos(1/2x)

• f(x) = cos(2x)

Multiple Choice

What is the equation of the graph?

•  y = 2 cos x
• y = sin x
• y = 2 sin x
• y = sin 2x 

Multiple Choice

What is the equation of the graph?

• y = 2sin(2x)

• y = 2cos(2x)

• y = 3sin(2x) - 1

• y = 3cos(2x) - 1

​We will also need to identify these things when looking at an equation.

Multiple Choice

$$y=\ 3\cos\left(2x\right)+4$$   3 represents the: 

• b

• amplitude

• phase shift

• midline

Multiple Choice

$$y=\ 3\cos\left(2x\right)+4$$   ,

2  represents the: 

• the number of units from the midline to the max

• amplitude

• b,

  number of cycles in 2pi

• midline

  y =

Multiple Choice

$$y=\ 3\cos\left(x-\pi\right)+4$$   $$4$$   represents the: 

• b

• amplitude

• horizontal  shift

• midline, vertical shift

Multiple Choice

How many cycles are there in 2pi or 360 degrees?

y = 3sin (7x) -2

• 7

• -2

• 3

• 6

Multiple Choice

What is the period of

y = 4cos(5x)

$$period\ =\ \frac{2\pi}{b}$$

• 2π/5

• π

• 5

• 5π

Multiple Choice

Write an equation of a sine function with

a period of π and amplitude of 7.

• y = 7sin(x) + 2

• y = 7sin(πx)

• y = 2sin(7x)

• y = 7sin(2x)

Multiple Choice

Which equation?

• y= (1/2)cos(3x)

• y= (1/2)sin(3x)

• y= cos(3x)

• y= (1/2)cos(2x)

Multiple Choice

which equation?

• y = 5 cos(2x) +4

• y = 5 sin(2x) +4

• y = 4 sin(2x) +5

• y = 9 sin(2x) + 4

Multiple Choice

What is the equation of graph? 

• y = sin 5x

• y =  cos 5x

• y = 5 sin x

• y = 5 cos x 

Multiple Choice

Which equation?

• y= sin(1/3x) − 1

• y= cos(3x) − 1

• y= cos(1/3x) − 1

• y= sin(3x) − 1

Multiple Choice

Which equation?

• y= sin(1/3x) − 1

• y= cos(3x) − 1

• y= cos(1/3x) − 1

• y= sin(3x) − 1