Questions: If θ = 1π/6, then find exact values for the following: sec (θ) equals □ csc (θ) equals □ tan (θ) equals □ cot (θ) equals □

Solution Steps

The given angle is \( \theta = \frac{\pi}{6} \).

The secant function is the reciprocal of the cosine function: \[ \sec(\theta) = \frac{1}{\cos(\theta)}. \] For \( \theta = \frac{\pi}{6} \), we know: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}. \] Thus: \[ \sec\left(\frac{\pi}{6}\right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}. \]

The cosecant function is the reciprocal of the sine function: \[ \csc(\theta) = \frac{1}{\sin(\theta)}. \] For \( \theta = \frac{\pi}{6} \), we know: \[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. \] Thus: \[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2. \]

The tangent function is the ratio of the sine function to the cosine function: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}. \] For \( \theta = \frac{\pi}{6} \), we know: \[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}. \] Thus: \[ \tan\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}. \]

Final Answer