Question

If $\tan \theta +\sec \theta =4$ , then what is the value of $\sin \theta$ ?

• A. $8/17$
• B. $8$/15
• C. $15/17$
• D. $23/32$

Answer 1

The correct option is C $15/17$

$\tan \theta +\sec \theta =4\Rightarrow \frac{1+\sin \theta }{\cos \theta }=4\Rightarrow \frac{1+\sin \theta }{4}=\cos \theta$

Also, we know that,

${\cos }^2\theta =1-{\sin }^2\theta \Rightarrow (\frac{1+\sin \theta }{4}{)}^2=1-{\sin }^2\theta \Rightarrow 1+{\sin }^2\theta +2\sin \theta =16-16{\sin }^2\theta \Rightarrow 17{\sin }^2\theta +2\sin \theta -15=0\Rightarrow (\sin \theta +1\left)\right(17\sin \theta -15)=0\Rightarrow \sin \theta =\frac{15}{17}or-1$

Therefore, Answer is $\frac{15}{17}$