Question

If cos theta = -2/3, which of the following are possible?

A. csc theta = 3/root5 and tan theta = -root5/2
B. sin theta = -root5/3 and tan theta = root5/2
C. sin theta = root5/3 and tan theta = root5/2
D. csc theta = -3/2 and tan that root 5/2

Answer 1

B.
`\sin \theta=(-√(5))/(3)` and
`\tan \theta =(√(5))/(2)`

Explanation:

We are given that,

`\cos \theta=(-2)/(3)`.

Since, we know,

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

i.e.
`\sin^2 \theta=1-\cos^2 \theta`

i.e.
`\sin^2 \theta=1-((-2)/(3))^2`

i.e.
`\sin^2 \theta=1-(4)/(9)`

i.e.
`\sin^2 \theta=(9-4)/(9)`

i.e.
`\sin^2 \theta=(5)/(9)`

i.e.
`\sin \theta=\pm (√(5))/(3)`

Also, we get,

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

i.e.
`\tan \theta =(\pm (√(5))/(3))/((-2)/(3))`

i.e.
`\tan \theta =\mp (√(5))/(2)`

So, we get that,

Option B is correct.