The Unit Circle

The Unit Circle

What goes around, comes around

​What is a Quadrant?

Each of the 4 quarters of a circle

• Quadrants of a circle are numbered

  going counter clockwise​

• The quadrant determines whether

  the x and y values are positive

  or negative​

Radians vs Degrees

REMEMBER

• (1, 0)

• (0, 1)

• (-1, 0)

• (0, -1)

Common Points on 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 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 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 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

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)

• $$-\frac{\sqrt{2}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$-\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

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

• $$-\frac{\sqrt{3}}{2}$$

• $$\frac{1}{2}$$

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

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  
  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$