Question

If sin theta = -5/7 , which of the following are possible?

A) cos theta = √24/7 , and tan theta = 5/√24
B) cos theta = -√24/7, and tan theta = 5/√24
C) sec theta = 7/√24 , and tan theta = -5/√24
D)sec theta = -7/5, and tan theta = 5/√24

Answer 1

Solution:

we are given that

$sin \theta=-5/7  $

As we know that

$cos \theta =\sqrt{1-sin^2 \theta}  $

Substitute the value we get

$cos \theta=\sqrt{1-(\frac{-5}{7})^2}  \\ \\ cos \theta=\sqrt{1-(\frac{25}{49})}  \\ \\ cos \theta=\sqrt{(\frac{24}{49})} =\frac{\sqrt{24}}{7}\\  $

Since $sec \theta=\frac{1}{cos \theta}\\  $

So here $sec \theta=\frac{7}{\sqrt{24}}\\  $

As we know that

$tan \theta=\frac{sin \theta}{cos \theta}\\ \\  \text{Substitute the values we get}\\ \\ tan \theta=\frac{-5/7}{\frac{\sqrt{24}}{7}}\\ \\ tan \theta=\frac{-5}{\sqrt{24}}\\  $

Hence the correct option is C.

Answer 2

It is given that

$sin \theta = \frac{-5}{7}$

And according to pythagorean identity, we will get

$cos \theta = \pm \sqrt{1-sin^2 \theta}$

Substituting the value of sin theta, we will get

$cos \theta = \pm \sqrt{1- \frac{25}{49}} \\ cos \theta = \pm \frac{\sqrt{24}}{7}$

and sec theta is the reciprocal of cos theta.

SO possible values of sec theta are

$sec \theta = \pm \frac{7}{ \sqrt{24}}$

And tan theta is the ratio of sin theta and cos theta

So possible values of tan theta are

$tan  \theta = \pm \frac{5}{ \sqrt{24}}$

So the correct options are B and C .