Question

which angle is coterminal with $-\frac{pi}{2}$?
$-\frac{7pi}{2}$
$\frac{3pi}{2}$
$\frac{5pi}{2}$

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles satisfy the formula \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\).

Step2: Substitute \(\theta =-\frac{\pi}{2}\) and \(k = - 1\) into the formula

$$ -\frac{\pi}{2}+2(-1)\pi=-\frac{\pi}{2}-2\pi=-\frac{\pi + 4\pi}{2}=-\frac{5\pi}{2} $$

Step3: Substitute \(\theta =-\frac{\pi}{2}\) and \(k = 1\) into the formula

$$ -\frac{\pi}{2}+2(1)\pi=-\frac{\pi}{2}+2\pi=\frac{- \pi+4\pi}{2}=\frac{3\pi}{2} $$

Step4: Substitute \(\theta =-\frac{\pi}{2}\) and \(k = 3\) into the formula

$$ -\frac{\pi}{2}+2(3)\pi=-\frac{\pi}{2}+6\pi=\frac{-\pi + 12\pi}{2}=\frac{11\pi}{2} $$

Step5: Substitute \(\theta =-\frac{\pi}{2}\) and \(k = - 3\) into the formula

$$ -\frac{\pi}{2}+2(-3)\pi=-\frac{\pi}{2}-6\pi=\frac{-\pi-12\pi}{2}=-\frac{13\pi}{2} $$

Among the given options, when \(k = 1\), \(-\frac{\pi}{2}+2\pi=\frac{3\pi}{2}\)

Answer:

\(\frac{3\pi}{2}\)