Previous If $\theta=\frac{-5 \pi}{6}$, then find exact values for the following: $\sec (\theta)$ equals $\csc (\theta)$ equals $\tan (\theta)$ equals $\cot (\theta)$ equals Question Help: $\square$ Video
Solution
Step 1 ：Given that the angle \(\theta = \frac{-5 \pi}{6}\), we need to find the exact values of the secant, cosecant, tangent, and cotangent of this angle.
Step 2 ：First, we find the values of cosine and sine for this angle. The angle \(\frac{-5 \pi}{6}\) is in the third quadrant, where both sine and cosine are negative.
Step 3 ：The reference angle for \(\frac{-5 \pi}{6}\) is \(\frac{\pi}{6}\), for which we know the sine and cosine values are \(\frac{1}{2}\) and \(\frac{\sqrt{3}}{2}\) respectively.
Step 4 ：So, for \(\theta = \frac{-5 \pi}{6}\), the cosine value will be \(-\frac{\sqrt{3}}{2}\) and the sine value will be \(-\frac{1}{2}\).
Step 5 ：Then, we calculate the secant, cosecant, tangent, and cotangent values using these sine and cosine values. Secant is the reciprocal of cosine, cosecant is the reciprocal of sine, tangent is sine divided by cosine, and cotangent is cosine divided by sine.
Step 6 ：Thus, the exact values for the following are: \(\sec (\theta) = \boxed{-1.1547005383792515}\), \(\csc (\theta) = \boxed{-2.0000000000000004}\), \(\tan (\theta) = \boxed{0.5773502691896256}\), and \(\cot (\theta) = \boxed{1.7320508075688776}\).
