Question

Find the exact values of the five remaining trig functions for B if sin(B) = 6/11.

Answer 1

First, recall the definition of the remaining trigonometric functions in terms of sin(x) and cos(x):

$\begin{gathered} \tan(x)=\frac{\sin(x)}{\cos(x)} \\ \\ \sec(x)=\frac{1}{\cos(x)} \\ \\ \csc(x)=\frac{1}{\sin(x)}} \\ \\ \cot(x)=\frac{\cos(x)}{\sin(x)} \end{gathered}$

On te other hand, recall the Pythagorean Identity:

$\sin^2(x)+\cos^2(x)=1$

We can obtain an expression for cos(x) in terms of sin(x) using the Pythagorean Identity as follows:

$\cos(x)=\sqrt{1-\sin^2(x)}$

Find cos(B) using the expression for cosine in terms of sine. Then, use the values of cos(B) andsin(B) to find the values of tan(B), sec(B), csc(B) and cot(B):

$\begin{gathered} \sin(B)=\frac{6}{11} \\ \\ \cos(B)=\sqrt{1-\left(\frac{6}{11}\right)^2}=\sqrt{1-\frac{36}{121}}=\sqrt{\frac{121-36}{121}}=\frac{\sqrt{85}}{11} \end{gathered}$

Then:

$\begin{gathered} \tan(B)=\dfrac{\frac{6}{11}}{\frac{\sqrt{85}}{11}}=\frac{6}{\sqrt{85}}=\frac{6\sqrt{85}}{85} \\ \\ \sec(B)=\dfrac{1}{\frac{\sqrt{85}}{11}}=\frac{11}{\sqrt{85}}=\frac{11\sqrt{85}}{85} \\ \\ \csc(B)=\frac{1}{\frac{6}{11}}=\frac{11}{6} \\ \\ \cot(B)=\frac{\frac{\sqrt{85}}{11}}{\frac{6}{11}}=\frac{\sqrt{85}}{6} \end{gathered}$

Therefore, the answers are:

$\begin{gathered} \sin(B)=\frac{6}{11} \\ \\ \cos(B)=\frac{\sqrt{85}}{11} \\ \\ \tan(B)=\frac{6\sqrt{85}}{85} \\ \\ \sec(B)=\frac{11\sqrt{85}}{85} \\ \\ \csc(B)=\frac{11}{6} \\ \\ \cot(B)=\frac{\sqrt{85}}{6} \end{gathered}$