Educational

Evaluate sec⁻¹(sec −7π/3) Evaluate \( \sec^{-1}(\sec -\frac{7\pi}{3}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec -\frac{7\pi}{3}\right) \] Step 1: Use identity \[ \sec(-x) = \sec x \] \[ \sec\left(-\frac{7\pi}{3}\right) = \sec\left(\frac{7\pi}{3}\right) \] Step 2: Reduce the angle \[ \frac{7\pi}{3} = 2\pi + \frac{\pi}{3} \] \[ \sec\left(\frac{7\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) \] Step 3: Apply inverse secant […]

Evaluate sec⁻¹(sec 9π/5) Evaluate \( \sec^{-1}(\sec \frac{9\pi}{5}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec \frac{9\pi}{5}\right) \] Step 1: Reduce the angle \[ \frac{9\pi}{5} = 2\pi – \frac{\pi}{5} \] \[ \sec\left(\frac{9\pi}{5}\right) = \sec\left(2\pi – \frac{\pi}{5}\right) = \sec\left(\frac{\pi}{5}\right) \] Step 2: Convert to cosine \[ \sec \frac{\pi}{5} = \frac{1}{\cos \frac{\pi}{5}} \] So we find angle

Evaluate sec⁻¹(sec 7π/3) Evaluate \( \sec^{-1}(\sec \frac{7\pi}{3}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec \frac{7\pi}{3}\right) \] Step 1: Reduce the angle \[ \frac{7\pi}{3} = 2\pi + \frac{\pi}{3} \] \[ \sec\left(\frac{7\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) \] Step 2: Convert to cosine \[ \sec \frac{\pi}{3} = 2 \Rightarrow \cos \theta = \frac{1}{2} \] Step 3: Apply inverse

Evaluate sec⁻¹(sec 5π/4) Evaluate \( \sec^{-1}(\sec \frac{5\pi}{4}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec \frac{5\pi}{4}\right) \] Step 1: Convert to cosine \[ \sec x = \frac{1}{\cos x} \] \[ \cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \Rightarrow \sec \frac{5\pi}{4} = -\sqrt{2} \] Step 2: Apply inverse secant \[ \sec^{-1}(-\sqrt{2}) \] Step 3: Use principal value range

Evaluate sec⁻¹(sec 2π/3) Evaluate \( \sec^{-1}(\sec \frac{2\pi}{3}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec \frac{2\pi}{3}\right) \] Step 1: Convert to cosine \[ \sec x = \frac{1}{\cos x} \] \[ \cos \frac{2\pi}{3} = -\frac{1}{2} \Rightarrow \sec \frac{2\pi}{3} = -2 \] Step 2: Apply inverse secant \[ \sec^{-1}(-2) \] Step 3: Use principal value range

Evaluate sec⁻¹(sec π/3) Evaluate \( \sec^{-1}(\sec \frac{\pi}{3}) \) Step-by-Step Solution We need to evaluate: \[ \sec^{-1}\left(\sec \frac{\pi}{3}\right) \] Step 1: Convert to cosine \[ \sec x = \frac{1}{\cos x} \] \[ \sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} = \frac{1}{\frac{1}{2}} = 2 \] Step 2: Apply inverse secant \[ \sec^{-1}(2) \] Step 3: Use principal value range

Evaluate tan⁻¹(tan 12) Evaluate \( \tan^{-1}(\tan 12) \) Step-by-Step Solution We need to evaluate: \[ \tan^{-1}(\tan 12) \] Step 1: Principal value range The principal value range of \( \tan^{-1}x \) is: \[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] Step 2: Reduce the angle Use periodicity \( \tan(x + \pi) = \tan x \) \[ 12 – 3\pi

Evaluate tan⁻¹(tan 4) Evaluate \( \tan^{-1}(\tan 4) \) Step-by-Step Solution We need to evaluate: \[ \tan^{-1}(\tan 4) \] Step 1: Principal value range The principal value range of \( \tan^{-1}x \) is: \[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] Step 2: Locate 4 Since \( 4 \in (\pi, \frac{3\pi}{2}) \), we adjust using periodicity: \[ \tan(x – \pi)

Evaluate tan⁻¹(tan 2) Evaluate \( \tan^{-1}(\tan 2) \) Step-by-Step Solution We need to evaluate: \[ \tan^{-1}(\tan 2) \] Step 1: Principal value range The principal value range of \( \tan^{-1}x \) is: \[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] Step 2: Check the angle Since \( 2 > \frac{\pi}{2} \), it lies outside the principal range. Step 3:

Evaluate tan⁻¹(tan 1) Evaluate \( \tan^{-1}(\tan 1) \) Step-by-Step Solution We need to evaluate: \[ \tan^{-1}(\tan 1) \] Step 1: Principal value range The principal value range of \( \tan^{-1}x \) is: \[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] Step 2: Check the angle Since \( 1 \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), it already lies in the principal range.