Mathematics: 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

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

julie47d

15.01.2022

B. $\sin \theta=\frac{-\sqrt{5}}{3}$ and $\tan \theta =\frac{\sqrt{5}}{2}$

Step-by-step explanation:

We are given that,

$\cos \theta=\frac{-2}{3}$ .

Since, we know,

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

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

i.e. $\sin^2 \theta=1-(\frac{-2}{3})^2$

i.e. $\sin^2 \theta=1-\frac{4}{9}$

i.e. $\sin^2 \theta=\frac{9-4}{9}$

i.e. $\sin^2 \theta=\frac{5}{9}$

i.e. $\sin \theta=\pm \frac{\sqrt{5}}{3}$

Also, we get,

$\tan \theta =\frac{\sin \theta}{\cos \theta}$

i.e. $\tan \theta =\frac{\pm \frac{\sqrt{5}}{3}}{\frac{-2}{3}}$

i.e. $\tan \theta =\mp \frac{\sqrt{5}}{2}$

So, we get that,

Option B is correct.

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