Question

Consider the right triangle shown below where a=8.09, b=9.4, and c=12.4. Note that θ and ϕ are measured in radians.What is the value of cos(θ)?cos(θ)= What is the value of sin(θ)?sin(θ)=What is the value of tan(θ)?tan(θ)=  What is the value of θ?θ=

Answer 1

By definition

`\cos (angle)=\frac{\text{ adjacent side}}{\text{ hipotenuse}}`

From the picture

`\begin{gathered} \cos (\theta)=(a)/(c) \\ \cos (\theta)=(8.09)/(12.4) \\ \cos (\theta)=0.65 \end{gathered}`

By definition

`\sin (angle)=\frac{\text{ opposite side}}{\text{ hipotenuse}}`

From the picture:

`\begin{gathered} \sin (\theta)=(b)/(c) \\ \sin (\theta)=(9.4)/(12.4) \\ \sin (\theta)=0.76 \end{gathered}`

By definition

`\tan (angle)=\frac{\text{ opposite side}}{\text{ adjacent side}}`

From the picture

`\begin{gathered} \tan (\theta)=(b)/(a) \\ \tan (\theta)=(9.4)/(8.09) \\ \tan (\theta)=1.16 \end{gathered}`

Isolating θ from the previous equations:

`\begin{gathered} \theta=\arccos (0.65)=49.46\text{ \degree}\approx49\text{ \degree} \\ \theta=\arcsin (0.76)=49.46\text{ \degree}\approx49\text{ \degree} \\ \theta=\arctan (1.16)=49.24\text{ \degree}\approx49\text{ \degree} \end{gathered}`

(The difference between the values is caused by rounding errors)