Question

I kinda forgot my trig (kinda)

full explanations required, (including citation of the identity)
3 questions

1.
`if`
`tan \theta= (12)/(5)` and
`0 \leq \theta \leq (\pi)/(2)`, then secΘ=
2.
`if`
`sin \theta= (3)/(5)` and
`0 \leq \theta \leq (\pi)/(2)`, then
`tan \theta=`
3. if
`cos \theta= (4)/(5)` and
`0 \leq \theta \leq (\pi)/(2)`, then
`cot \theta=`

show all work

Answer 1

:p irlya its 81

Explanation:

Answer 2

`\sec^2\theta=1+\tan^2\theta\implies \sec\theta=\pm√(1+\tan^2\theta)`

Since
`\cos\theta>0` for
`0<\theta<\frac\pi2`, we have
`\sec\theta>0`, so we take the positive root. Now,


`\sec\theta=\sqrt{1+\left(\frac{12}5\right)^2}=\frac{13}5`

- - -


`\tan\theta=(\sin\theta)/(\cos\theta)`

`\cos^2\theta=1-\sin^2\theta`

In the first quadrant, cosine is positive, so


`\cos\theta=√(1-\sin^2\theta)`

and in turn,


`\sin\theta=\frac35\implies\cos\theta=√(1-\left(\frac35\right)^2)=\frac45`

`\implies\tan\theta=(\frac35)/(\frac45)=\frac34`

- - -

In the previous problem, we had
`\cos\theta=\frac45`, so we must have
`\tan\theta=\frac34`, which means


`\cot\theta=\frac1{\tan\theta}=\frac1{\frac34}=\frac43`