How do you find the value of sin^(-1)(sin (cos^(-1) (sin (pi/12))))?

2 Answers
Ratnaker Mehta
Jul 17, 2016

sin^-1(sin(cos^-1(sin(pi/12))))=5pi/12.

Explanation:

We will use the following Rules :

(R1) : sintheta=cos(pi/2-theta)

(R2) : cos^-1(costheta)=theta, theta in [o,pi]

(R3) : sin^-1(sintheta)=theta, theta in [-pi/2,pi/2]

Let us note that, by

(R1), sin(pi/12)=cos(pi/2-pi/12)=cos(5pi/12)

So, cos^-1(sin(pi/12))=cos^-1(cos(5pi/12)), where,

5pi/12 in [0,pi], so, using (R2), we get,

cos^-1(cos(5pi/12))=5pi/12

Again, as 5pi/12 in [-pi/2,pi/2], by (R3), we have,

sin^-1(sin(5pi/12))=5pi/12

Finally, sin^-1(sin(cos^-1(sin(pi/12))))=5pi/12.

Hope, this will be of Help! Enjoy Maths.!

A. S. Adikesavan
Jul 17, 2016

(5pi)/12=75^o

Explanation:

Use sin a = cos (pi/2-a) and,

if y=f(x) , x = f^(-1)y, f f^(-1)y=y and f^(-1)yf(x)=x.

The given expression is

sin^(-1)sin cos^(-1)sin(pi/12)

=sin^(-1)sin (cos^(-1)cos(pi/2-pi/12))

=sin^(-1)sin((5pi)/12)

=(5pi)/12

Related questions
See all questions in Trigonometric Functions of Any Angle
Impact of this question
2400 views around the world
You can reuse this answer
Creative Commons License