Unit Circle Trig Practice

Evaluating Sine & Cosine with the Unit Circle

Goal: Students will be able to evaluate sine and cosine using the Unit Circle

Using the Unit Circle

• to evaluate sine and cosine, we use the coordinates that lie on the Unit Circle's circumference

• cosine of an angle is equal to the length of the reference triangle's adjacent side and is represented by the x-coordinate

• sine of an angle is equal to the length of the reference triangle's opposite side and is represented by the y-coordinate

Evaluating Cosine

• cos(θ) = x-coordinate of the point associated with that angle

• cos(270°) = 0

• cos(π/6) = √3/2

• cos(2π/3) = -1/2

Multiple Choice

Evaluate cos(135°)

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

Multiple Choice

Evaluate cos(11π/6)

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

Fill in the Blanks

Evaluate cos(90°)

Evaluating Sine

• sin(θ) = y-coordinate of the point associated with that angle

• sin(4π/3) = -√3/2

• sin(0) = 0

• sin(135°) = √2/2

Multiple Choice

Evaluate sin(60°)

• 0

• $$\frac{1}{2}$$  
  

• $$\frac{\sqrt{2}}{2}$$  
  

• $$\frac{\sqrt{3}}{2}$$  

• 1

Multiple Choice

sin(7π/6)

• - ½

• √3 /2

• ½

• -√3 /2

• -√2 /2

Multiple Choice

sin 180°

• 1

• -1

• 0

• $$-\frac{\sqrt{3}}{2}$$

• $$\frac{1}{2}$$

The Other Trigonometric Functions

Values of cotangent, secant and cosecant

Multiple Choice

REVIEW

What is sin 30°?

• ^√3/₂

• 1

• ½

• ^√2/₂

Multiple Choice

REVIEW

What is cos 90°?

• ½

• undefined

• 1

• 0

Multiple Choice

REVIEW

What is sin 45°?

• 1

• ^√3/₂

• ½

• ^√2/₂

Multiple Choice

REVIEW

What is cos 60°?

• ½

• ^√3/₂

• ^√2/₂

• 1

The TANGENT (TOA) Function - A reminder

• The TANGENT of an angle is defined in Geometry by the ratio of the opposite side and adjacent side.

• The TANGENT of an angle is defined by the UNIT CIRCLE as y/x.

• The TANGENT of an angle in Trigonometry is defined by the ratio of sinΘ and cosΘ

Multiple Choice

The GEOMETRY use of TANGENT;

Find the trig value. Take note you are looking for the tangent of angle C.

• A

• B

• C

• D

Multiple Choice

One - more:

Find the trig value. Again, stating the tangent of C.

• A

• B

• C

• D

The Trigonometry Use

The tangent is equivalent to the sin value divided by the cosine value. Which is the same idea as y divided by cosine? Right??

Multiple Choice

Which of the following is the value of;

(It might be helpful to convert to a degree first.)

$$\tan\left(\frac{7\pi}{4}\right)$$

• -1

• 1

• √2/2

• undefined

Multiple Choice

Which of the following is the value of;

(It might be helpful to convert to a degree first.)

$$\tan\left(-\pi\right)$$

• undefined

• 0

• 1

• -1

Multiple Choice

Which of the following is the value of;

(It might be helpful to convert to a degree first.)

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

• √3/3

• √3

• $$-\frac{\sqrt{3}}{3}$$  

• $$-\sqrt{3}$$  

The Reciprocal Functions

• The three main trigonometric function values are ratios and ratios have reciprocals.

• The reciprocal of sine is COSECANT (CSC).

• The reciprocal of cosine is SECANT (SEC)

• The reciprocal of tangent is COTANGENT (COT)

So if they are reciprocals of one another

• If sinA = 3/5 then cscA = 5/3

• If cosA = 4/5 then secA =5/4

• If tanA = 3/4 then cotA = 4/3

Multiple Choice

Let's practice some reciprocals...

Given $$\frac{5}{9}$$ , the reciprocal would be ___.

• $$\frac{9}{5}$$

• $$\frac{5}{9}$$

• Neither of these

Multiple Choice

Let's practice some reciprocals...

Given $$\frac{15}{7}$$ , the reciprocal would be ___.

• $$\frac{7}{15}$$

• $$\frac{15}{7}$$

• Neither of these

Multiple Choice

How about this?

Given $$\frac{\pi}{180}$$ , the reciprocal would be ___.

• $$\frac{180}{\pi}$$

• $$\frac{\pi}{180}$$

• Neither of these

Consider this then;

Multiple Choice

What is the reciprocal of the sine (sin) function?

• cosecant (csc)

• cosine (cos)

• secant (sec)

• cotangent (cot)

Multiple Choice

Cotangent (cot) is the reciprocal of what function?

• Cosecant (csc)

• Sine (sin)

• Cosine (cos)

• Tangent (tan)

Multiple Choice

What is the reciprocal of the cosine (cos) function?

• cosecant (csc)

• cosine (cos)

• secant (sec)

• cotangent (cot)

Multiple Choice

Which of the following is the exact value of;

 csc(30^o).

(Remember csc is the reciprocal of sin.)

• 2

• $$\frac{\sqrt{3}}{3}$$

• $$\frac{2\sqrt{3}}{3}$$

• $$\sqrt{2}$$

• undefined

Multiple Choice

Which of the following is the exact value of;

 cot(135^o)

(Remember...cot is the reciprocal of tan)

• 1

• -1

• 2

• $$\sqrt{2}$$

• undefined

Multiple Choice

Which of the following is the exact value of;  $$\csc\left(\frac{\pi}{2}\right)$$  

(Remember, csc is the reciprocal of sin)

• -1

• $$\frac{\sqrt{3}}{3}$$

• 1

• $$\sqrt{2}$$

• undefined

Multiple Choice

Which of the following is the exact value of;

  $$\cot\left(\pi\right)$$  

(Remember, cot is the reciprocal of tan)

• -2

• $$\frac{\sqrt{3}}{3}$$

• $$\frac{2\sqrt{3}}{3}$$

• $$\sqrt{2}$$

• undefined

Multiple Choice

Not all angles are exact value angles. Consider this.

Given $$\cos\theta=-\frac{2}{3}$$ the $$\sec\theta$$ =?

• $$-\frac{2}{3}$$

• $$-\frac{3}{2}$$

• $$\frac{3}{2}$$

• $$\frac{2}{3}$$