2. if $csc\theta =-\frac{4sqrt{6}}{3}$, find $sin\theta$.
2. if $csc\theta =-\frac{4sqrt{6}}{3}$, find $sin\theta$.
Answer
Explanation:
Step1: Recall the reciprocal identity
The reciprocal identity for cosecant and sine is (\csc\theta=\frac{1}{\sin\theta}).
Step2: Solve for (\sin\theta)
If (\csc\theta =-\frac{4\sqrt{6}}{3}), then (\sin\theta=\frac{1}{\csc\theta}). Substitute the value of (\csc\theta) into the formula: (\sin\theta=\frac{1}{-\frac{4\sqrt{6}}{3}}). Simplify the expression: (\sin\theta =-\frac{3}{4\sqrt{6}}). Rationalize the denominator by multiplying the numerator and denominator by (\sqrt{6}): (\sin\theta=-\frac{3\sqrt{6}}{4\times6}=-\frac{\sqrt{6}}{8}).
Answer:
(-\frac{\sqrt{6}}{8})