Question

Find the exact value of each of the remaining trigonometric functions of $\theta$

$\sin \theta =\frac{-2}{3},\mathrm{\pi }<\mathrm{\theta }<\frac{3\mathrm{\pi }}{2}$

Short answer

The values are

$\cos \theta =\frac{-\sqrt{5}}{3},csc\theta =\frac{-3}{2},sec\theta =\frac{-3}{\sqrt{5}},\tan \theta =\frac{2}{\sqrt{5}},cot\theta =\frac{\sqrt{5}}{2}$

Step-by-step solution

.Given information

$\sin \theta =\frac{-2}{3},\mathrm{\pi }<\mathrm{\theta }<\frac{3\mathrm{\pi }}{2}$is given.

we have to find the exact value of each of the remaining trigonometric functions

Step 2.Finding the value of cosθ

$$\begin{gathered} Since{\sin }^2\theta +{\cos }^2\theta =1,here\sin \theta =\frac{-2}{3} \\ \cos \theta =\pm \sqrt{1-{\sin }^2\theta },here\theta lieson\mathrm{\pi }<\theta <\frac{3\mathrm{\pi }}{2}so\sin \theta isonthirdquadrantandallotherexcept\tan \theta andcot\theta arewillbenegative. \\ hence\cos \theta =\sqrt{1-{\sin }^2\theta } \\ =-\sqrt{1-\frac{{(-2)}^2}{3^2}} \\ =-\sqrt{1-\frac{4}{9}} \\ =-\sqrt{\frac{5}{9}} \\ =\frac{-\sqrt{5}}{3} \end{gathered}$$

Step 3. Find the values ofcscθ and  secθ

$$\begin{gathered} \mathrm{In}\mathrm{this}\mathrm{question}\mathrm{csc\theta }=\frac{1}{\mathrm{sin\theta }} \\ =\frac{1}{{\displaystyle -\frac{2}{3}}},\mathrm{sin\theta }=-\frac{2}{3} \\ =-\frac{3}{2} \\ \mathrm{cos\theta }=\frac{-\sqrt{5}}{3},\mathrm{then}\mathrm{sec\theta }=\frac{1}{\mathrm{cos\theta }} \\ =\frac{1}{{\displaystyle \frac{-\sqrt{5}}{3}}} \\ =\frac{-3}{\sqrt{5}} \end{gathered}$$

Step 4. Finding the value of tan θ and cotθ

$$\begin{gathered} weknowthat\tan \theta =\frac{\sin \theta }{\cos \theta },\sin \theta =\frac{-2}{3}and\cos \theta =\frac{-\sqrt{5}}{3} \\ =\frac{{\displaystyle \frac{{\displaystyle -2}}{{\displaystyle 3}}}}{{\displaystyle \frac{{\displaystyle -\sqrt{5}}}{{\displaystyle 3}}}} \\ =\frac{2}{\sqrt{5}} \\ cot\theta =\frac{1}{\tan \theta } \\ =\frac{1}{{\displaystyle \frac{2}{\sqrt{5}}}} \\ =\frac{\sqrt{}}{2} \end{gathered}$$