Question

If sin-1x+sin-1y=2pi/3 then find the value of cos-1x+cos-1y

Answer 1

sin-1x+sin-1y=2π/3
we know that, sin-1x+cos-1x=π/2; or cos-1x=π/2-sin-1x
∴, cos-1x+cos-1y=π/2-sin-1x+π/2-sin-1y=(π/2+π/2)-(sin-1x+sin-1y)=π-2π/3
=(3π-2π)/3=π/3

Answer 2

The value of $\cos -1 x+\cos -1 y \text { is } \frac{\pi}{3}$.

To find:

Find the value of cos-1x + cos-1y

Solution:

Given: $\sin -1 x+\sin -1 y=\frac{2 \pi}{3}$

We know that $\sin -1 x+\cos -1 x=\frac{\pi}{2} \text { or } \cos -1 x=\frac{\pi}{2}-\sin -1 x$

Hence,  

$\cos ^{-1} x+\cos ^{-1} y$

$\begin{aligned} &=\frac{\pi}{2}-\sin -1 x+\frac{\pi}{2}-\sin -1 y \\ \\ &=\left(\frac{\pi}{2}+\frac{\pi}{2}\right)-(\sin -1 x+\sin -1 y) \\ \\ &=\pi-\frac{2 \pi}{3} \\ \\ &=\frac{3 \pi-2 \pi}{3} \\ \\ &=\frac{\pi}{3} \end{aligned}$

Sine, cosine and tangent are main functions in trigonometry. We are mainly derived this functions using formulas. ‘Trigonometric functions’ have been ‘extended as functions’ of a “real or complex variable”, which are today ‘pervasive in all mathematics’.