Question

a(1) cos1sim1 -2

Answer 1

The inverse of a function f:A→B exists if f is one-one onto i.e.,

y=f( x )⇒ f −1 ( y )=x .

The given inverse trigonometry function is tan −1 ( 1 )+ cos −1 ( − 1 2 )+ sin −1 ( − 1 2 ) .

Let,

tan −1 ( 1 )=x tanx=1 =tan π 4

Therefore, tan −1 ( 1 )= π 4

Let,

cos −1 ( − 1 2 )=y

cosy=− 1 2 =−cos( π 3 ) =cos( π− π 3 ) =cos( 2π 3 )

Therefore, cos −1 ( − 1 2 )= 2π 3 .

Let,

sin −1 ( − 1 2 )=z

sinz=− 1 2 =−sin( π 6 ) =sin( − π 6 )

According to the question, summation of all the functions gives,

tan −1 ( 1 )+ cos −1 ( − 1 2 )+ sin −1 ( − 1 2 ) = π 4 + 2π 3 − π 6 = 3π+8π−2π 12 = 9π 12 = 3π 4

Thus, the value of tan −1 ( 1 )+ cos −1 ( − 1 2 )+ sin −1 ( − 1 2 ) is 3π 4 .