Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value of each of the remaining trigonometric functions of $\mathrm{\theta }.$

$\mathrm{sec\theta }=2,\mathrm{sin\theta }<0.$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions are,

$$\begin{gathered} \mathrm{sin\theta }=-\frac{\sqrt{3}}{2} \\ \mathrm{tan\theta }=-\sqrt{3} \\ \mathrm{cot\theta }=-\frac{\sqrt{3}}{3} \\ \mathrm{csc\theta }=-\frac{2\sqrt{3}}{3} \end{gathered}$$

Step 1. Given Information

Given that the function is $\mathrm{sec\theta }=2,\mathrm{sin\theta }<0.$ To find the exact values using substitution.

Step 2. Phythagorean identity

It implies that $\mathrm{cos\theta }=\frac{1}{\mathrm{sec\theta }}=\frac{1}{2}$. Since $\mathrm{sin\theta }<0,\mathrm{cos\theta }=\frac{1}{2}>0.$ On using Phythagorean identity as ${\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }=1$. On using this equation to solve for $\mathrm{sin\theta },$

$$\begin{gathered} {\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }=1 \\ \mathrm{sin\theta }=\pm \sqrt{1-{\cos }^2\mathrm{\theta }} \end{gathered}$$

On substituting the values,

$$\begin{gathered} \mathrm{sin\theta }=-\sqrt{1-{\left(\frac{1}{2}\right)}^2} \\ =-\sqrt{1-\frac{1}{4}} \\ =-\sqrt{\frac{3}{4}} \\ \mathrm{sin\theta }=-\frac{\sqrt{3}}{2} \end{gathered}$$

Step 3. Definition of unit circle

The definition of unit circle is $\mathrm{sin\theta }=\mathrm{y},\mathrm{cos\theta }=\mathrm{x}.$

$$\begin{gathered} \mathrm{tan\theta }=\frac{\mathrm{y}}{\mathrm{x}} \\ \mathrm{tan\theta }=\frac{\mathrm{sin\theta }}{\mathrm{cos\theta }} \\ =\frac{-{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{2}}} \\ \mathrm{tan\theta }=-\sqrt{3} \\ \mathrm{cot\theta }=\frac{1}{\mathrm{tan\theta }} \\ =\frac{1}{-\sqrt{3}} \\ \mathrm{cot\theta }=-\frac{\sqrt{3}}{2} \\ \mathrm{csc\theta }=\frac{1}{\mathrm{sin\theta }} \\ =-\frac{{\displaystyle \frac{1}{\sqrt{3}}}}{2} \\ =-\frac{2}{\sqrt{3}} \\ \mathrm{csc\theta }=-\frac{2\sqrt{3}}{3} \end{gathered}$$