Question

Given cos theta=4/7 and csc theta<0, find sin theta and tan theta.

Answer 1

C.
`\sin\theta =-(√(33))/(7)` and
`\tan\theta=(-√(33))/(4)}`

Explanation:

We have that,
`\cos \theta =(4)/(7)`.

As, it is given that,
`\sin^(2)\theta +\cos^(2)\theta=1`

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

i.e.
`\sin^(2)\theta =1-((4)/(7))^(2)`

i.e.
`\sin^(2)\theta =1-(16)/(49)`

i.e.
`\sin^(2)\theta =(49-16)/(49)`

i.e.
`\sin^(2)\theta =(33)/(49)`

i.e.
`\sin\theta =\pm (√(33))/(7)`

Thus, according to options,
`\sin\theta =-(√(33))/(7)`

Now, as we know that,

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

i.e.
`\tan\theta=(-(√(33))/(7))/((4)/(7))`

i.e.
`\tan\theta=(-√(33))/(4)}`

Hence, option C is correct.