Question

If tan(θ) = − 8/ 9 and sin(θ) > 0, find the exact values of each of the trig functions

how would you work this out? could someone please explain?

Answer 1

\[
\tan ^2(\theta
)+1=\left(\frac{8}{9}\right
)^2+1=\sec
^2(\theta )

\]
Solve and you find
\[

\sec(\theta)=\pm\frac{\sqrt{145}}{9}

\]
Which sign do you choose?

Answer 2

You choose the negative sign since \( \cos(\theta)<0\)

Answer 3

\[\Large
\cos(\theta)=-\frac{9}{\sqrt {145}}
\]

Answer 4

Can you find the remaining?

Answer 5

i dont know how to find them. havent done these questions before

Answer 6

Solve this to find \(\sin(\theta)\)
\[

\tan (\theta )=\frac{\sin
(\theta )}{\cos (\theta
)}=-\frac{8}{9}=-\frac{\sin
(\theta
)}{\frac{9}{\sqrt{145}}} \\ \Large
\sin
(\theta )=\frac{8}{\sqrt{145}}\\ \Large
\csc(\theta)=\frac{\sqrt{145} }{8}
\]

Answer 7

\[
\Large \cot(\theta)=-\frac 98

\]