4.2 Trigonometric Functions: The Unit Circle

Presentation

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Mathematics

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11th Grade

•

Medium

•

CCSS

HSF.TF.A.2, HSF.TF.A.4, HSF.TF.A.1

Standards-aligned

Larry Cooper

Used 7+ times

FREE Resource

12 Slides • 15 Questions

4.2 Trigonometric Functions: 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.

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

Multiple Choice

What is cos 60°? #1

^√3/₂

^√2/₂

Multiple Choice

What is sin 45°? #2

^√3/₂

^√2/₂

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

Multiple Choice

What is tan 45°? #5

^√2/₂

^√3/₂

Here is an overview of all major angles in radians on the unit circle:

Multiple Choice

Evaluate $\tan\left(\frac{7\pi}{6}\right)$ . #14

$-\frac{\sqrt{3}}{2}$

$\frac{\sqrt{3}}{3}$

$-\frac{1}{2}$

$\sqrt{3}$

Multiple Choice

Evaluate $\sin\left(\frac{23\pi}{6}\right)$ . #15

$-\frac{\sqrt{3}}{2}$

$\frac{\sqrt{3}}{3}$

$-\frac{1}{2}$

$\sqrt{3}$

Multiple Choice

Find the equivalent value for $\cot\ -300\degree.$ #19

$-\cot60\degree$

$\frac{1}{\tan\ 60\degree}$

$\frac{1}{\tan300\degree}$

$\cot120\degree$

Multiple Select

Which of the following is a point on the unit circle? #20

(Check all that apply)

$\left(1,\ 1\right)$

$\left(-1,\ 0\right)$

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

$\left(\frac{1}{\sqrt{5}},-\frac{2}{\sqrt{5}}\right)$

Multiple Choice

What is the correct ordered pair for an angle rotation of $150\degree$ ? #21

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

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

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

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

Multiple Choice

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

(Multiply by $\frac{180}{\pi}$ to get degrees) #37

(−√2∕2, √2∕2)

(−√2∕2, −√2∕2)

(−√3∕2, −√2∕2)

(√2∕2, −√2∕2)

Match

Match the following angles measured in radians to degrees. #41

$90\degree$

$180\degree$

$30\degree$

$45\degree$

$60\degree$

$\frac{\pi}{2}$

$\pi$

$\frac{\pi}{6}$

$\frac{\pi}{4}$

$\frac{\pi}{3}$

Match

Match the correct value of the trigonometric ratios of special angles. #42

$\frac{\sqrt[]{2}}{2}$

$\frac{\sqrt[]{3}}{2}$

$\frac{\sqrt[]{3}}{3}$

$\frac{1}{2}$

$\sin\frac{\pi}{4}$

$\cos\frac{\pi}{6}$

$\tan\frac{\pi}{6}$

$\cos\frac{\pi}{3}$

$\sin\frac{\pi}{6}$

Multiple Choice

What is $\sec\left(\frac{7\pi}{4}\right)$ ? #7

$\frac{\sqrt{2}}{2}$

$\sqrt{2}$

$-\frac{\sqrt{2}}{2}$

$-\sqrt{2}$

Multiple Choice

Find the sin(-9π/4). #8

-√3/2

-1/2

√2/2

-√2/2

Multiple Choice

What is the tan(7π/6)? #9

_√3

-√3/3

√3/3

Multiple Choice

What is the csc (7π/6)? #10

-2

-1/2

1/2

4.2 Trigonometric Functions: The Unit Circle

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