Given:
Right triangle
To find:
The six trigonometric functions of θ
Solution:
Hypotenuse = 18
Adjacent side to θ = 10
Opposite side to θ = ?
Using Pythagoras theorem:
[tex]\text {Hypotenuse}^2 = \text{adjacent}^2+\text{opposite}^2[/tex]
[tex]18^2 =10^2+\text{opposite}^2[/tex]
[tex]324=100+\text{opposite}^2[/tex]
Subtract 100 from both sides.
[tex]224=\text{opposite}^2[/tex]
Taking square root on both sides.
[tex]4\sqrt{14}=\text{opposite}[/tex]
Using trigonometric ratio formula:
[tex]$\sin\theta =\frac{\text{opposite }}{\text{hypotenuse}}[/tex]
[tex]$\sin\theta =\frac{4\sqrt{14} }{18}[/tex]
[tex]$\csc\theta =\frac{\text{hypotenuse}}{\text{opposite }}[/tex]
[tex]$\csc\theta =\frac{18}{4\sqrt{14} }[/tex]
[tex]$\cos \theta=\frac{\text { adjacent side }}{\text { hypotenuse }}[/tex]
[tex]$\cos \theta=\frac{10}{18}[/tex]
[tex]$\sec \theta=\frac{\text { hypotenuse }}{\text { adjacent side }}[/tex]
[tex]$\sec \theta=\frac{18 }{10}[/tex]
[tex]$\tan \theta=\frac{\text { opposite side }}{\text { adjacent side }}[/tex]
[tex]$\tan \theta=\frac{4\sqrt{14} }{10}[/tex]
[tex]$\cot \theta=\frac{\text { adjacent side }}{\text { opposite side }}[/tex]
[tex]$\cot \theta=\frac{10}{4\sqrt{14} }[/tex]
