Question

If \(tan\ \theta=\sqrt{3}\), then the value of \(sec\ θ\) is

• A. 2
• B. \(\frac{1}{2}\)
• C. \(\frac{\sqrt{3}}{2}\)
• D. \(\frac{2}{\sqrt{3}}\)

Answer 1

The Correct Option is A

Given: \( \tan \theta = \sqrt{3} \)

Step 1: Express in terms of sin and cos 

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \sqrt{3} \]

Step 2: Assign standard values

For \( \tan \theta = \sqrt{3} \), we know: \[ \sin \theta = \frac{\sqrt{3}}{2}, \quad \cos \theta = \frac{1}{2} \]

Step 3: Compute sec θ

\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{2}} = 2 \]

Final Answer: 2

Answer 2

If \( \tan \theta = \sqrt{3} \), we need to find the value of \( \sec \theta \).

We know from trigonometric identities that:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)

Given that \(\tan \theta = \sqrt{3}\), we identify this with the standard angle \(\theta = 60^\circ\) (or \(\frac{\pi}{3}\) radians) because \(\tan 60^\circ = \sqrt{3}\).

At this angle, the trigonometric values are:

• \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 60^\circ = \frac{1}{2}\)

Subsequently, using the identity for secant:

\(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{2}} = 2\)

Thus, the value of \(\sec \theta\) is 2.