Answer:
From the given ratio for [tex]tan\theta=\frac{4}{5}[/tex] we found the values are
[tex]sin\theta=4[/tex]
[tex]cos\theta=5[/tex]
[tex]csc\theta=\frac{1}{4}[/tex]
[tex]sec\theta=\frac{1}{5}[/tex]
[tex]cot\theta=\frac{5}{4}}[/tex]
Step-by-step explanation:
Given the ratio for [tex]tan\theta=\frac{4}{5}[/tex]
We have to find the rest of trignometric functions:
[tex]tan\theta=\frac{4}{5}\hfill (1)[/tex]
[tex]tan\theta[/tex] can be written as
[tex]tan\theta=\frac{sin\theta}{cos\theta}\hfill (2)[/tex]
Comparing equations (1) and (2) we get
[tex]tan\theta=\frac{sin\theta}{cos\theta}=\frac{4}{5}[/tex]
[tex]\frac{sin\theta}{cos\theta}=\frac{4}{5}[/tex]
Equating the coressponding numerator and denominator respectively
Therefore [tex]sin\theta=4[/tex] and [tex]cos\theta=5[/tex]
We can find [tex]csc\theta[/tex]
[tex]csc\theta=\frac{1}{sin\theta}[/tex]
[tex]=\frac{1}{4}[/tex] (since [tex]sin\theta=4[/tex])
[tex]csc\theta=\frac{1}{4}[/tex]
We can find [tex]sec\theta[/tex]
[tex]sec\theta=\frac{1}{cos\theta}[/tex]
[tex]=\frac{1}{5}[/tex] (since [tex]cos\theta=5[/tex])
[tex]sec\theta=\frac{1}{5}[/tex]
To find [tex]cot\theta[/tex]:
[tex]cot\theta=\frac{1}{tan\theta}[/tex]
[tex]=\frac{1}{\frac{4}{5}}[/tex]
[tex]=1\times \frac{5}{4}[/tex]
[tex]=\frac{5}{4}}[/tex]
[tex]cot\theta=\frac{5}{4}}[/tex]
