Using the Unit Circle

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.

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

Multiple Choice

Evaluate cos(225°)

• $-\frac{\sqrt{2}}{2}$

• $\frac{\sqrt{2}}{2}$

• $-\frac{\sqrt{3}}{2}$

• $-\frac{1}{2}$

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

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

• $$-\frac{1}{2}$$

• $$\frac{1}{2}$$

• $$\frac{\sqrt{3}}{2}$$

• $$-1$$

• $$1$$

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

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°)

• $-\frac{\sqrt{2}}{2}$

• $\frac{\sqrt{2}}{2}$

• $-\frac{\sqrt{3}}{2}$

• $-\frac{1}{2}$