Questions: If θ=-23π/6, then find exact values for the following:
csc (θ) equals
Transcript text: If $\theta=-\frac{23 \pi}{6}$, then find exact values for the following:
$\csc (\theta)$ equals
Solution
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.
Solution Approach
Normalize the angle \(\theta = -\frac{23\pi}{6}\) to an equivalent angle between \(0\) and \(2\pi\).
Calculate the sine of the normalized angle.
Take the reciprocal of the sine value to get \(\csc(\theta)\).
Step 1: Normalize the Angle
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:
\[
\theta_{\text{normalized}} = 0.5236
\]
Step 2: Calculate the Sine of the Normalized Angle
Next, we calculate the sine of the normalized angle:
\[
\sin(0.5236) \approx 0.5000
\]
Step 3: Calculate the Cosecant
The cosecant is the reciprocal of the sine:
\[
\csc(\theta) = \frac{1}{\sin(\theta_{\text{normalized}})} = \frac{1}{0.5000} \approx 2.0000
\]