Questions: If θ=-23π/6, then find exact values for the following: csc (θ) equals

Solution Steps

To find the exact value of \(\csc(\theta)\) for \(\theta = -\frac{23\pi}{6}\), we need to first find the equivalent angle in the standard position (between \(0\) and \(2\pi\)). Then, we can find the sine of that angle and take its reciprocal to get the cosecant.

1. Normalize the angle \(\theta = -\frac{23\pi}{6}\) to an equivalent angle between \(0\) and \(2\pi\).
2. Calculate the sine of the normalized angle.
3. Take the reciprocal of the sine value to get \(\csc(\theta)\).

To find the equivalent angle of \(\theta = -\frac{23\pi}{6}\) within the interval \([0, 2\pi)\), we use the modulo operation: \[ \theta \mod 2\pi = -\frac{23\pi}{6} \mod 2\pi \] Given the output, the normalized angle is: \[ \theta_{\text{normalized}} = 0.5236 \]

Next, we calculate the sine of the normalized angle: \[ \sin(0.5236) \approx 0.5000 \]

The cosecant is the reciprocal of the sine: \[ \csc(\theta) = \frac{1}{\sin(\theta_{\text{normalized}})} = \frac{1}{0.5000} \approx 2.0000 \]

Final Answer