Question

If \(\sin\theta=\dfrac{\sqrt{3}}{2}\), then the value of \(\csc\theta+\cot\theta\) is

• A. \(2+\sqrt{3}\)
• B. \(2\sqrt{3}\)
• C. \(\sqrt{2}\)
• D. \(\sqrt{3}\)

Hint:

When \(\sin\theta\) or \(\cos\theta\) is standard, compute others using \(\sin^{2}\theta+\cos^{2}\theta=1\).

Answer 1

The Correct Option is D

Step 1: Compute co-functions.
\(\csc\theta=\dfrac{1}{\sin\theta}=\dfrac{2}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3}\).
\(\cos\theta=\sqrt{1-\sin^{2}\theta}=\dfrac{1}{2}\Rightarrow \cot\theta=\dfrac{\cos\theta}{\sin\theta}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}\).
Step 2: Add.
\(\csc\theta+\cot\theta=\dfrac{2\sqrt{3}}{3}+\dfrac{\sqrt{3}}{3}=\sqrt{3}\).