Question

If sin $\theta =\frac{-4}{5}$ and $\pi <0<\frac{3\pi }{2}$, find the values of all the other five trigonometric functions.

Answer 1

Clearly, $\theta$ lies in the third quadrant in which tan $\theta$ and cot $\theta$ are positive and all the other trigonometric functions are negative.

Now, sin $\theta =\frac{4}{5}\Rightarrow cosec\theta =\frac{1}{sin\theta }=\frac{-5}{4}$

$\therefore co{t}^2\theta =(cose{c}^2\theta -1)=(\frac{25}{16}-1)=\frac{9}{16}$

$\Rightarrow cot\theta =+\sqrt{\frac{9}{16}}=\frac{3}{4}$

$\Rightarrow tan\theta =\frac{1}{cot\theta }=\frac{4}{3}$

$\text{Also, cos}\theta =\text{cot}\theta sin\theta =\left(\frac{3}{4}\right)\times \left(\frac{-4}{5}\right)=\frac{-3}{5}$

$\therefore sec\theta =\frac{1}{cos\theta }=\frac{-5}{3}$ $\text{Hence,}\text{cos}\theta =\frac{-3}{5};\text{tan}\theta =\frac{4}{3};cot\theta =\frac{3}{4};$

$\text{sec}\theta =\frac{-5}{3}\text{and cosec}\theta =\frac{-5}{4}$