Given that
[tex]\sin \theta=\frac{\sqrt[]{11}}{6}[/tex]
Required: Value of csc θ
Solution:
Step 1:
From the reciprocal trigonometric identities,
[tex]\csc \theta=\frac{1}{\sin\theta}\text{ ----- equation 1}[/tex]
where
[tex]\sin \theta=\frac{\sqrt[]{11}}{6}[/tex]
Step 2:
Substitute the value of sin θ into equation 1.
Thus,
[tex]\begin{gathered} \csc \theta=\frac{1}{\sin\theta} \\ \csc \theta=\frac{1}{\frac{\sqrt[]{11}}{6}}=\frac{6}{\sqrt[]{11}} \end{gathered}[/tex]
Step 3:
Rationalize the surd obtained in step 2.
Thus, we have
[tex]\begin{gathered} \csc \theta=\frac{6}{\sqrt[]{11}} \\ \text{Multiply the numerator and denominator by }\sqrt[]{11.} \\ \text{thus,} \\ \Rightarrow\frac{6}{\sqrt[]{11}}\times\frac{\sqrt[]{11}}{\sqrt[]{11}} \\ =\frac{6\sqrt[]{11}}{11} \end{gathered}[/tex]
Hence, the value of csc θ is evaluated to be
[tex]\frac{6\sqrt[]{11}}{11}[/tex]
