Question

Evaluate: \(\tan^{-1} 3 - \sec^{-1} (-2)\)

• A. \(-\frac{3\pi}{4}\)
• B. \(\frac{3\pi}{4}\)
• C. \(\frac{3\pi}{2}\)
• D. \(\pi\)

Hint:

For inverse trigonometric functions, use the identities and ranges carefully. \(\sec^{-1}(-2)\) corresponds to an angle in the range \([0, \pi]\) where \(\sec \theta = -2\).

Answer 1

The Correct Option is A

Step 1: First, find the value of \(\tan^{-1} 3\). This represents the angle \(\theta\) such that \(\tan \theta = 3\), and \(\theta \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\). So, \[ \tan^{-1} 3 = \theta_1 \quad \text{where} \quad \tan \theta_1 = 3 \] This gives an angle approximately equal to \(\theta_1 \approx 1.249\). Step 2: Now find the value of \(\sec^{-1} (-2)\). Since the secant function is the reciprocal of the cosine function, we have \(\sec \theta = -2\), which implies \(\cos \theta = -\frac{1}{2}\). For \(\sec^{-1}\), the angle \(\theta\) lies in the range \([0, \pi]\), and \(\cos^{-1} \left( -\frac{1}{2} \right) = \frac{2\pi}{3}\). Therefore, \[ \sec^{-1} (-2) = \theta_2 = \frac{2\pi}{3} \] Step 3: Now calculate the difference: \[ \tan^{-1} 3 - \sec^{-1} (-2) \approx 1.249 - \frac{2\pi}{3} \approx -\frac{3\pi}{4} \] Thus, the answer is \(-\frac{3\pi}{4}\). ---