Question

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

$\cos \theta =\frac{4}{5},{270}^o<\theta <{360}^o$

Short answer

The exact values are

$\sin \theta =-\frac{3}{5},sec\theta =\frac{5}{4},csc\theta =-\frac{5}{3},\tan \theta =-\frac{3}{4},cot\theta =-\frac{4}{3}$

Step-by-step solution

Step 1. Given information

$\cos \theta =\frac{4}{5},{270}^0<\theta <{360}^0$is given.

We have to find the exact value of each of the remaining trigonometric functions.

Step 2. Find the value of sinθ

$$\begin{gathered} Weknowthat{\sin }^2\theta +{\cos }^2\theta =1,here\cos \theta =\frac{4}{5} \\ \sin \theta =\pm \sqrt{1-{\cos }^2\theta },here\theta lieson{270}^o<\theta <{360}^{o}so\cos \theta isonfourthquadrantallotherexcept\cos \theta andsec\theta arewillbenegative. \\ hence\sin \theta =-\sqrt{1-co{s}^2\theta } \\ =-\sqrt{1-\frac{4^2}{5^2}} \\ =-\sqrt{1-\frac{16}{25}} \\ =-\frac{3}{5} \end{gathered}$$

 step 3. Find the values of secθ, cscθ

According to the question

$$\begin{gathered} \cos \theta =\frac{4}{5},thensec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{{\displaystyle \frac{4}{5}}} \\ =\frac{5}{4} \\ csc\theta =\frac{1}{\sin \theta } \\ =\frac{1}{{\displaystyle \frac{-3}{5}}} \\ =\frac{-5}{3} \end{gathered}$$

Step 4. Find the values of tanθ and cotθ

$$\begin{gathered} Since\tan \theta =\frac{\sin \theta }{\cos \theta },\sin \theta =\frac{-3}{5}and\cos \theta =\frac{4}{5} \\ =\frac{{\displaystyle \frac{-3}{5}}}{{\displaystyle \frac{4}{5}}} \\ =-\frac{3}{4} \\ cot\theta =\frac{1}{\tan \theta } \\ =\frac{1}{{\displaystyle -\frac{3}{4}}} \\ =-\frac{4}{3} \end{gathered}$$