AP Calc Unit Circle Refresher

Used to organize the stars and give them coordinates

Trig functions originated in trig

​

​Astronomy

Used to determine geometric patterns in buildings

Curving surfaces (like glass or steel)

Finding Heights

Creating 3-D figures​

Architecture

Real World Applications

Some text here about the topic of discussion

​Real World Applications Continued...

The unit circle has helped with digital imaging, such as computer graphics, voice to text, or photography.

Also, it helped predict movements of water, light, and sound.​

Using the Unit Circle

Goal: Students will be able to evaluate sine, cosine, and tangent using the unit circle.

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.

Radians vs Degrees

​What is a Radian?

• ​s is the length of the intercepted arc

• r is the radi​us of the circle

​Converting Radians to Degrees

Converting ​Degrees to Radians

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

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

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.

Multiple Choice

What is the standard equation of a circle?

• (x-k)² + (y-h)² = r²

• (x+h)² + (y+k)² = r²

• (x-h)² + (y-k)² = r²

• (k-x)² + (h-y)² = r²

Multiple Choice

What does (h,k) standard for in the standard circle equation?

• (h,k) is the center of the circle

• (h,k) is a point on the circle.

• (h,k) is a point that we guess and find

• (h,k) is the distance of the circle

Multiple Choice

The unit circle...

• Has center at (1,1)

• Has a circumference of 1

• Has a diameter of 1

• Has a radius of 1

Multiple Choice

The formula for the unit circle is

• $$a^2+b^2=c^2$$  

• $$x^2+y^2=1$$  

• $$y=mx+b$$  

• $$x+y=1$$  

• $$\sin+\cos=\tan$$  

Multiple Choice

Angles are measured in which direction from standard position?

• Clockwise

• Counterclockwise

Multiple Choice

The coordinates of the points on the unit circle are written in which order?

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

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

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

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

tanθ 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

Tangent is...

• a. Opposite/hypotenuse

• b. sine/cosine

• c. neither a or b

• d. both a and b

Multiple Choice

What quadrant is -2π/3 in?

• I

• II

• III

• IV

Multiple Choice

Evaluate cos(225°)

• $$-\frac{\sqrt{2}}{2}$$

• $$\frac{\sqrt{2}}{2}$$

• $$-\frac{\sqrt{3}}{2}$$

• $$-\frac{1}{2}$$

Multiple Choice

Evaluate  $$\sin\left(\frac{\pi}{2}\right)$$  

• $$-\frac{1}{2}$$

• $$\frac{1}{2}$$

• $$\frac{\sqrt{3}}{2}$$

• $$-1$$

• $$1$$

Multiple Choice

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

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$  

Multiple Choice

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

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$  

Multiple Choice

Evaluate cos(-135°)

• $$-\frac{\sqrt{2}}{2}$$

• $$\frac{\sqrt{2}}{2}$$

• $$-\frac{\sqrt{3}}{2}$$

• $$-\frac{1}{2}$$

Multiple Choice

Identify the Positive angle in both radians and degrees that represent point D.

• $$60\degree,\ \frac{\pi}{3}$$

• $$45\degree,\ \frac{\pi}{4}$$

• $$-60\degree,\ -\frac{\pi}{3}$$

• $$90\degree,\ \frac{\pi}{2}$$

Multiple Choice

Identify the Positive angle in both radians and degrees that represent point L.

• $$240\degree,\ \frac{4\pi}{3}$$

• $$225\degree,\ \frac{5\pi}{4}$$

• $$-60\degree,\ -\frac{\pi}{3}$$

• $$270\degree,\ \frac{3\pi}{2}$$

Multiple Choice

Identify the Positive angle in both radians and degrees that represent point I.

• $$180\degree,\ \pi$$

• $$210\degree,\ \frac{7\pi}{6}$$

• $$150\degree,\ \frac{5\pi}{6}$$

• $$360\degree,\ 2\pi$$

Multiple Choice

Find the exact value of the trig ratio. $$\cos30\degree$$   

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{1}{2}$$  

Multiple Choice

Find the exact value of the trig ratio. $$\sin180\degree$$   

• $$0$$  

• $$1$$  

• $$\frac{1}{2}$$

• $$-1$$  

Multiple Choice

Find the exact value of the trig ratio. $$\cos\left(\frac{5\pi}{3}\right)$$   

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{1}{2}$$

• $$-\frac{\sqrt{3}}{2}$$  

Multiple Choice

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

• (−√2∕2, √2∕2)

• (−√2∕2, −√2∕2)

• (−√3∕2, −√2∕2)

• (√2∕2, −√2∕2)

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

sec 3π/2

• 0

• undefined

• 1

• -1

Multiple Choice

tan 7π/4

• -1

• 1

• √2/2

• undefined

Multiple Choice

cos 4π/3

• -√3/2

• √3/2

• 1/2

• -1/2

Multiple Choice

tan (-π)

• undefined

• 0

• 1

• -1

Multiple Choice

Convert the following to degrees:
-8π/5 radians

• -288⁰

• -260⁰

• -275⁰

Multiple Choice

What is 7π/4 radians in degrees?

• 300

• 135

• 315

• 45

Multiple Choice

Convert 150⁰ to radians

• 5π/6

• 3π/4

• 7π/6

• 2π/3

Multiple Choice

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

• (−√2∕2, √2∕2)

• (−√2∕2, −√2∕2)

• (−√3∕2, −√2∕2)

• (√2∕2, −√2∕2)

Multiple Choice

Find cos(600°)

• (-1/2)

• (-√3/2)

• (√3/2)

• (√2/2)

Multiple Choice

Find sin(-240°)

• (√3)/2)

• (-1/2)

• (√2/2)

• (1/2)

Multiple Choice

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

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$  

Multiple Choice

cot 90

• 0

• 1

• -1

• undefined

Multiple Choice

sec 135

• √2

• -√2

• √2/2

• -√2/2