Precalculus Final Review

Match

Now, use the given triangle to represent $$\sin\theta$$ and $$\cos\theta$$ in terms of the variables given in the diagram.

• $$\frac{y}{1}=y$$

• $$\frac{x}{1}=x$$

• 1

sinθ

cosθ

OP

• sinθ

• cosθ

• OP

Dropdown

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



Reference angle = ​

Select answer

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

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

The unit circle is a great tool for finding the exact value of trig functions, but it is not your only tool. Sometimes it is still convenient to think of using right triangles.

Fill in the Blank

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.

Find the missing side length.

Dropdown

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.



Given that the missing side has length 3,

$$\sin\theta=$$ ​

Select answer

Multiple Choice

sin (π/2)

• √3/2

• √2/2

• 1/2

• 1

Multiple Choice

cos (π/4)

• √3/2

• √2/2

• 1/2

• 1

Multiple Choice

cos(π)

• √3/2

• -√2/2

• -1

• 1

Multiple Choice

cos(5π/3)

• -√2/2

• -1/2

• 1/2

• √3/2

Multiple Choice

cos(30^o)

• - ½

• √3/2

• 2√3 /3

• ½

Multiple Choice

Choose the best match for the graph.

• 30^o

• 45^o

• 60^o

• 90^o

Multiple Choice

Choose the best match for the graph.

• 30^o

• 45^o

• 60^o

• 90^o

Multiple Choice

Based on the unit circle, what is the value of sin?

• y

• x

• x/y

• y/x

Multiple Choice

Based on the unit circle, what is the value of cos?

• y

• x

• x/y

• y/x

Multiple Choice

sin(11π/6)

• -1/2

• 1/2

• √3/2

• -√3/2

Multiple Choice

How many radians does a circle have?

•   360 degrees

• 2pi radians

• pi radians

• ^{2 radians}

Multiple Choice

Convert π radians to degrees.

• 90⁰

• 180⁰

• 360⁰

• 60⁰

Multiple Choice

Convert 150⁰ to radians

• 5π/6

• 3π/4

• 7π/6

• 2π/3

Multiple Choice

Convert 100⁰ to radians

• 5π/9

• π/2

• 5π/8

• 5π/6

Multiple Choice

What is 7π/4 radians in degrees?

• 300

• 135

• 315

• 45

Multiple Choice

What is 225  degrees in radians? 

• 7π/4

• 5π/4

• 3π/4

• 5π/3

Multiple Choice

Find the missing side lengths.

• a = 4, b = 4

• a = 4, b = 2√2

• a = 2√2,  b = 4

• a = 4, b = 4

Multiple Choice

What is the length of x in this 45-45-90 triangle?

• 4√2

• 4

• 8

• 4√3

Multiple Choice

What is the length of u and v in this 30-60-90 triangle?

• u = 8 v = 4√3

• u = 4√2 v = 8

• u = 16 v = 8

• u = 4√3 v = 8

Multiple Choice

Find a.

• 3

• $$3\sqrt{3}$$

• 6

• $$\sqrt{3}$$

Multiple Choice

A central angle of a circle measures  $$230\degree$$  and its radius is 6 cm. What is the length of the arc intercepted by the angle?

• 12.04 cm

• 24.08 cm

• 144.51 cm

• 72.26 cm

Multiple Choice

What is the radius of a circle that has a central angle of  $$230\degree$$  and an arc length of 60 miles?

• 6.10 miles

• 29.89 miles

• 74.95 miles

• 14.95 miles

Multiple Choice

What is angular velocity

• the rate of angular displacement of an object

• the resistance of an object to change its current state

• the amount of angular motion of an object

Multiple Choice

How many radians make up one revolution of 360 degrees?

• π/2

• π

• 2π

• 3π/2

Multiple Choice

What are the units for angular velocity

• m/s2

• rad/s

• kgm/s

• kg/m2

Multiple Choice

A disk rotates with an angular speed $$\omega$$  . What is the linear speed v of a point on the disk located a distance r away from the axis of rotation?

• $$\omega r$$  

• $$\frac{\omega}{r}$$  

• $$\frac{r}{\omega}$$  

• $$\omega^2r$$  

Multiple Choice

A record spins at 33 rpm (revolutions per minute), which is an angular velocity of about 3.46 radians per second. What is the approximate linear velocity of a fly that sits on the record, 12 cm from the center?

• 3.4 cm/s

• 21.5 cm/s

• 41.5 cm/s

• 131.9 cm/s

Multiple Choice

Klyde's angular velocity is

• the same as Bonnie's

• twice Bonnie's

• half of Bonnie's

• 1/4 Bonnie's

Multiple Choice

Which letter represents the point

$$(-4,-\frac{2π}{3})$$  ?

• B

• F

• D

• A

Multiple Choice

Which letter represents the point

$$(-4,\frac{π}{2})$$  ?

• D

• C

• B

• A

Multiple Choice

Which point represents (-4, π/2)

• D

• C

• B

• A

Multiple Choice

Which letter represents the point

$$(5,π)$$  ?

• F

• E

• B

• A

Multiple Choice

How do you write an ordered pair in polar form?

• (x,y)

• (θ, r)

• (r,θ)

• (r,y)

Multiple Choice

When converting between rectangular and polar coordinates:

y = ?

• rcosθ

• rsinθ

• x² + y²

• y/x, x ≠ 0

Multiple Choice

10.1-10.2 Converting between rectangular and polar coordinates:
x = ?

• rcosθ

• rsinθ

• x² + y²

• y/x, x ≠ 0

Multiple Choice

Convert the polar coordinates to rectangular form

• ( -3, 3)

• (3, 3√3)

• (-3, 3√3)

• (3, -3√3)

Multiple Choice

Find polar coordinates for the point:
(√3 , 1)

• (2, π/6)

• (-2, -π/6)

• (1,0)

• (4π,-2)

Multiple Choice

Find the rectangular coordinates of each polar point. (4, 3π/2)

• (4,4)

• (4,0)

• (0,-4)

• (0,1)

Multiple Choice

Convert from rectangular to polar coordinates. (-4, 4)

• (4√2, π/4)

• (4√2, 3π/4)

• (4√2, -π/4)

• (-4√2, -7π/4)

Multiple Choice

Convert from polar coordinates, (6, 5π/4) to rectangular coordinates.

• (-3√2, -3√2)

• (3√2, 3√2)

• (6.0, 0.4)

• (-6.0, -0.4)

Multiple Choice

Convert the polar coordinates to rectangular coordinates.