Question

For any acute angle $\theta$ select the trigonometric functions with a positive value.

• A. $\sec \left(\frac{7\pi }{2}+\theta \right)$
• B. $\cos \left(\frac{5\pi }{2}-\theta \right)$
• C. $\mathrm{cosec}\left(\frac{5\pi }{2}+\theta \right)$
• D. $\sin \left(\frac{3\pi }{2}-\theta \right)$

Answer 1

Answer: (A), (B), (C)

$\mathrm{cosec}\left(\frac{5\pi }{2}+\theta \right)$

Given an acute angle $\theta$ and we have to find which of the following trigonometric functions have a positive value.

$a.\sin (\frac{3\pi }{2}-\theta )$
$\frac{3\pi }{2}-\theta$ corresponds to angle in the third quadrant where both $\sin$ and $\cos$ are negative.
$\Rightarrow \sin (\frac{3\pi }{2}-\theta )=-\cos \theta$
Hence, it's value will be negative.

$b.\cos (\frac{5\pi }{2}-\theta )$
$\frac{5\pi }{2}-\theta$ corresponds to an angle in the first quadrant where all trigonomettric functions are positive.
$\Rightarrow \cos (\frac{5\pi }{2}-\theta )=+\sin \theta$
Hence, it's value will be positive.

$c.\sec (\frac{7\pi }{2}+\theta )$
$\frac{7\pi }{2}+\theta$ corresponds to angle in the fourth qudrant where sec of an angle is positive..
$\Rightarrow \sec (\frac{7\pi }{2}+\theta )=\text{cosec}\theta$
Hence, it's value will be positive.

$c.\text{cosec}(\frac{5\pi }{2}+\theta )$
$\frac{5\pi }{2}+\theta$ corresponds to angle in the second qudrant where cosec of an angle is positive.
$\Rightarrow \text{cosec}(\frac{7\pi }{2}+\theta )=\sec \theta$
Hence, it's value will be positive.