Circles and Triangles

8.2: Special Right Triangles & Unit Circle

Objective: Use special right triangles and reference angles to create the Unit Circle.

Reference Angle

• A reference angle is an angle in standard position on the coordinate plane

• Standard Position: positive acute angle where initial side is on the positive x-axis and terminal side is in quadrant one.

How to find Reference Angles

• Quadrant 1: Angle Measure = Reference Angle

• Quadrant 2: 180 - Angle Measure

• Quadrant 3: Angle Measure - 180

• Quadrant 4: 360 - Angle Measure

Reference Angle Example

• Find the reference angle of

   $\theta=115\degree$  

• First, what quadrant is 115 in?

•  $115\degree$  is in Quadrant 2, so we know Reference Angle = 180 - Angle Measure

• Reference Angle =  $180\ -\ 115\ =\ 65\degree$  

• Reference Angle is  $65\degree$  

Coterminal Angle

• Coterminal Angles are angles with the same initial and terminal sides but have different measures.

•  $-240\degree,\ 120\degree,\ 480\degree,\ 840\degree...\$  are all coterminal angles.

• Do you see a pattern?

How to Find Coterminal Angles

• Degrees: To find coterminal angles, Coterminal Angles = Angle $\pm360\cdot n$  where  $n\$ is any integer

• Radians: To find coterminal angles, Coterminal Angles = Angle  $\pm2\pi\cdot n$  where  $n$  is any integer

Coterminal Angle Example

• Find a positive and negative coterminal angle of  $150\degree$  

• To find a positive coterminal angle:  $150+360\ =\ 510\degree$  

• To find a negative coterminal angle:  $150-360=-210\degree$  

Coterminal Angle Example

• Find a positive and negative of  $\frac{5\pi}{4}$  

• To find a positive coterminal angle:  $\frac{5\pi}{4}+2\pi=\frac{5\pi}{4}+\frac{8\pi}{4}=\frac{13\pi}{4}$  

• To find a negative coterminal angle:  $\frac{5\pi}{4}-2\pi=\frac{5\pi}{4}-\frac{8\pi}{4}=-\frac{3\pi}{4}$  

The Unit Circle

Khan Academy Video

The Unit Circle

•  $\left(x,\ y\right)=\left(\cos\left(\theta\right),\ \sin\left(\theta\right)\right)$  

•  $\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}=\frac{y}{x}$