Right Triangle Trigonometry

Right Triangle Trigonometry

Angles and Their Measures

An angle is formed when a ray is rotated around its endpoint.

Positive & Negative Angles

The ray in its original position is called the initial ray or the initial side of an angle. In the Cartesian plane, we assume the initial side of an angle is the positive x-axis. The ray after it is rotated is called the terminal ray or the terminal side of an angle. Rotation in a counterclockwise direction corresponds to a positive angle, whereas rotation in a clockwise direction corresponds to a negative angle. 

Positive & Negative Angles

Open Ended

Estimate the measure, in degrees, of the angle shown in the diagram.

Open Ended

Estimate the measure, in degrees, of the angle shown in the diagram.

Degree Measure of Angles 

One way to measure the size of an angle is with degree measure. An angle formed by one complete counterclockwise rotation has measure 360 degrees, denoted 360°. 

Right Angles

An angle measuring exactly 90° is called a right angle.

A right angle is often represented by

two rays that are perpendicular to each other. 

Straight Angles

An angle measuring exactly 180° is called a straight angle. 

Multiple Choice

Acute Angles

• $0\degree<\theta<90\degree$

• $\theta<90\degree$

• $0\degree\le\theta\le90\degree$

• $\theta=45\degree$

Multiple Choice

Obtuse Angles

•  $90\degree<\theta<180\degree$ 

•  $\theta<180\degree$ 

•  $0\degree<\theta<180\degree$ 

•  $\theta=150\degree$ 

Radians

In geometry and most everyday applications, angles are measured in degrees. However, in calculus a more natural angle measure is radian measure. 

Radians

A central angle is an angle that has its vertex at the center of a circle. When the intercepted arc’s length is equal to the radius, the measure of the central angle is 1 radian. 

Converting Between Degrees & Radians

To convert degrees to radians, multiply the degree measure (given) by  $\frac{\pi}{180\degree}$ .

Example: Convert  $45\degree$  to radians.

 $45\degree\cdot\frac{\pi}{180\degree}=\frac{45\degree\cdot\pi}{180\degree}=\frac{\pi}{4}$   

Converting Between Degrees & Radians

To convert radians to degrees, multiply the radian measure (given) by  $\frac{180\degree}{\pi}$ .

Example: Convert  $\frac{3\pi}{2}$  to degrees.

 $\frac{3\pi}{2}\cdot\frac{180\degree}{\pi}=\frac{3\cdot\pi\cdot180\degree}{2\cdot\pi}=270\degree$   

Poll

Convert  $65\degree$  to radians. 

•  $\frac{\pi}{3}$  

•  $\frac{13}{36}\pi$  

•  $\frac{36}{13}\pi$  

Poll

Convert  $-\frac{\pi}{2}$  in degrees. 

•  $90\degree$  

•  $-90\degree$  

•  $60\degree$  

Right Triangles & Basic Trigonometric Functions

•  $\sin\left(\alpha\right)$ :  "sine of angle alpha"

•  $\cos\left(\beta\right)$ : "cosine of angle beta" 

•  $\tan\left(\theta\right)$ : "tangent of angle theta" 

•  $\alpha,\ \beta,\ \theta,\ \gamma$  (Greek letters) are used to name angles

Evaluating the Basic Trig Functions

 $\sin\left(\theta\right)=\frac{opposite}{hypotenuse}$  

 $\cos\left(\theta\right)=\frac{adjacent}{hypotenuse}$  

 $\tan\left(\theta\right)=\frac{opposite}{adjacent}$  

SOH CAH TOA

SOH:          $\sin\left(\theta\right)=\frac{opposite}{hypotenuse}$ 

CAH:          $\cos\left(\theta\right)=\frac{adjacent}{hypotenuse}$  

TOA:          $\tan\left(\theta\right)=\frac{opposite}{adjacent}$  

Multiple Choice

What is the first step to solve for a?

• $\sin\left(56\right)=\frac{a}{15}$

• $\cos\left(56\right)=\frac{a}{15}$

• $\tan\left(56\right)=\frac{b}{a}$

• $\cos\left(56\right)=\frac{15}{6}$

Multiple Choice

The first step to solve for a was:  $\cos\left(56\right)=\frac{a}{15}$  . Which one is the second step? 

•  $a=15\cdot\cos\left(56\right)$  

•  $a=\frac{15}{\cos\left(56\right)}$  

•  $a=\frac{\cos\left(56\right)}{15}$  

•  $a=\cos\left(56\cdot15\right)$