Question

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

role="math" localid="1646521470980" $\mathrm{sin\theta }=\frac{2}{3},\mathrm{tan\theta }<0$

Short answer

The exact values are,

$$\begin{gathered} \mathrm{cos\theta }=-\frac{\sqrt{5}}{3} \\ \mathrm{tan\theta }=-\frac{2\sqrt{5}}{5} \\ \mathrm{csc\theta }=\frac{3}{2} \\ \mathrm{sec\theta }=-\frac{3\sqrt{5}}{5} \\ \mathrm{cot\theta }=-\frac{\sqrt{5}}{2} \end{gathered}$$

Step-by-step solution

Step 1. Given Information

Given that the function is $\mathrm{sin\theta }=\frac{2}{3},\mathrm{tan\theta }<0.$To find the exact values of $\mathrm{\theta }$.

Step 2. Using theorem and signs

There is a difference in sign due to the quadrant of II or IV. When $\mathrm{y}=2>0,$the quadrant is in II. The angle lies to the point $\mathrm{P}=(\mathrm{x},\mathrm{y})$in the standard position. The radius of the circle has ${\mathrm{r}}^2={\mathrm{x}}^2+{\mathrm{y}}^2.$The values are $\mathrm{sin\theta }=\frac{\mathrm{y}}{\mathrm{r}},\mathrm{y}=2,\mathrm{r}=3.$On solving and substituting,

$$\begin{gathered} {\mathrm{x}}^2={\mathrm{r}}^2-{\mathrm{y}}^2 \\ {\mathrm{x}}^2={\left(3\right)}^2-{\left(2\right)}^2{\mathrm{x}}^2=9-4 \\ {\mathrm{x}}^2=5 \\ \mathrm{x}=-\sqrt{5} \end{gathered}$$

Substitution to find exact values

On substituting the values for exact values are,

$$\begin{gathered} \mathrm{cos\theta }=\frac{\mathrm{x}}{\mathrm{r}} \\ \mathrm{cos\theta }=-\frac{\sqrt{5}}{3} \\ \mathrm{tan\theta }=\frac{\mathrm{y}}{\mathrm{x}} \\ \mathrm{tan\theta }=-\frac{2}{\sqrt{5}}.\frac{\sqrt{5}}{\sqrt{5}} \\ \mathrm{tan\theta }=-\frac{2\sqrt{5}}{5} \end{gathered}$$

Substitution to find exact values

On finding the exact values are,

$$\begin{gathered} \mathrm{csc\theta }=\frac{\mathrm{r}}{\mathrm{y}} \\ \mathrm{csc\theta }=\frac{3}{2} \\ \mathrm{sec\theta }=\frac{\mathrm{r}}{\mathrm{x}} \\ \mathrm{sec\theta }=-\frac{3}{\sqrt{5}}.\frac{\sqrt{5}}{\sqrt{5}} \\ \mathrm{sec\theta }=-\frac{3\sqrt{5}}{5} \\ \mathrm{cot\theta }=\frac{\mathrm{x}}{\mathrm{y}} \\ \mathrm{cot\theta }=-\frac{\sqrt{5}}{2} \end{gathered}$$