Question

Find the exact value of each real number y if it exists. Do not use a calculator.y = csc⁻¹ (―2)

Accepted Answer

Recall that the function $$ y = \csc^{-1}(x)  is $$the inverse cosecant function, which gives the angle $$ y $$ such that $$ \csc y = x . $$Here, we want to find $$ y $$ such that $$ \csc y = -2 . $$Rewrite the equation $$ \csc y = -2  in $$terms of sine, since $$ \csc y = \frac{1}{\sin y} . $$This gives $$ \frac{1}{\sin y} = -2 $$, so $$ \sin y = -\frac{1}{2} . $$Determine the general solutions for $$ y $$ where $$ \sin y = -\frac{1}{2} . $$Recall that sine is negative in the third and fourth quadrants, and the reference angle for $$ \sin y = \frac{1}{2}  is$$ $$ \frac{\pi}{6} . $$Write the specific solutions for $$ y  in $$the interval $$ [-\frac{\pi}{2}, \frac{\pi}{2}]  or $$the principal range of $$ \csc^{-1} $$, which is usually $$ [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] $$, excluding zero. Identify which of these solutions fall into the principal range. Express the exact value(s) of $$ y  in $$radians that satisfy the equation $$ y = \csc^{-1}(-2) $$ within the principal range, using the reference angle $$ \frac{\pi}{6} $$ and the appropriate sign.

Find the exact value of each real number y if it exists. Do not use a calculator.
y = csc⁻¹ (―2)

Verified video answer for a similar problem:

Key Concepts

Inverse Cosecant Function (csc⁻¹)

Relationship Between Cosecant and Sine

Exact Values of Sine for Special Angles