Topic 7 Trigonometric Functions

Topic 7 Trigonometric Functions

Sine and Cosine

Sine and Cosine are defined as:

Sine is the opposite side divided by the hypotenuse in a right triangle.

Cosine is the adjacent side divided by the hypotenuse in a right triangle.

Finding trigonometric values from one given trigonometric function

Given $\cos\phi=\frac{8}{10},\ find\ \cot\ and\ \csc\ \phi.$  

First, find the missing side:
 $a^2+8^2=10^2$  
 $a^2=100-64$ 
  $a^2=36$  
 $\sqrt{a^2}=\sqrt{36}$  
 $a=6$  

Find trigonometric values from one given trigonometric function

Remember,  $a=6,\ b=8,\ and\ c=10$ 
 $\cos\phi=\frac{8}{10},\ so\ \sin\phi=\frac{6}{10},\ and\ \tan\phi=\frac{6}{8}$    $\cot\phi=\frac{8}{6}$  

Open Ended

 $Given\ \sin\phi=\frac{8}{17},\$  find  $\sec\phi\ and\ \tan\phi$  

Graph of tangent

• The graph of tangent is increasing everywhere in its domain.

• The period is $\pi$ 

• It has asymptotes starting at  $\frac{\pi}{2}$  and every  $\frac{\pi+2x\pi}{2}where\ x=an\ integer.$  

Open Ended

 $What\ is\ the\ height\ of\ a\ building\ that\ cast\ a\ 40\ foot\ shadow\$   $when\ the\ earth\ is\ at\ a\ 60\degree\ to\ the\ sun?$  (Give an answer to the nearest foot)

Error Analysis

 Rodrigo was asked to identify the phase shift for the graph:

 $y=-\cos\left(\theta-\frac{\pi}{6}\right)-4$  
 $He\ said\ the\ the\ phase\ shift\ was\ \frac{\pi}{6}to\ the\ left$  and 4 units down.
Is he correct?

Multiple Choice

What is  $\cos\theta\ if\ \sin\theta=\frac{-9}{15}and\ \theta\ is\ in\ Quadrant\ IV$  

•  $\cos\theta=\frac{9}{15}$  

•  $\cos\theta=\frac{12}{15}$  

•  $\cos\theta=\frac{-12}{15}$  

•  $\cos\theta=\frac{-9}{15}$