Find the exact value of the expression sin1 squareroot 22 co

Find the exact value of the expression. sin^-1 squareroot 2/2 cos^-1 (-square squareroot 3/2) tan^-1 (-1) Use a calculator to find the value of the expression rounded to two decimal places, sin^-1 (0.7) Find the exact value of the expression. Do not use a calculator. Sin^-1 [sin (pi/5)] cos^-1 [cos (-3 pi/5)]

Solution

1 a) sin^-1(sqrt2/2)

let y =sin^-1(sqrt2/2)

siny = sqrt2/2

y = pi/4

So, sin^-1(sqrt2/2) = pi/4

b) cos^-1(-sqrt2/2)

y =cos^-1(-sqrt2/2)

cosy = -sqrt(2)/2

range of cos^-1x [0, pi]

So, y = pi -pi/4 = 3pi/4

So, cos^-1(-sqrt2/2) = 3pi/4

c) tan^-1(-1)

y = tan^-1(-1)

tany = -1

range of tan^-1x = [-pi/2 , pi/2]

So,y= -pi/4

2) sin^-1(0.7)

y = sin^-1(0.7)

siny = 0.7

range of sin^-1x [ -pi/2, pi/2]

So, y = 0.775

3)

a) sin^-1[sin(pi/5)]

y = sin^-1[sin(pi/5)]

siny = sin (pi/5)

y = pi/5

b) y = cos^-1[cos( -3pi/5)]

cosy = cos(-3pi/5)

y = -3pi/5