Find the exact value of each expression if it is defined a a

Find the exact value of each expression, if it is defined, (a) arcsin (-Squareroot 3/2) (b) arccos (- Squareroot 2/2) (c) arctan (- Squareroot 3) Find the exact value of each expression, if it is defined. (a) arcsin (sin(60 degree)) (b) cos(arccos (- Squareroot 2/2)) (c) arctan(tan (3 pi/4))

Solution

a) arcsin (-sqrt 3/2)

let arcsin (-sqrt 3/2) = x

sin x = -sqrt 3/2

Range of arcsinx [ -pi/2 , pi/2]

So, x = - pi/3 = -60 degrees

b) arc cos ( - sqrt 2/2 )

arc cos ( - sqrt 2/2 ) = x

cos x = - sqrt 2/2

Range of arccosx lies in interval [ 0, pi]

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

c) arctan ( -sqrt 3)

tan x = -sqrt 3

Range of arctanx [ -pi/2 , pi/2]

x = -pi/3

3) i) arcsin(sin60)

y =  arcsin(sin60)

siny = sin60

y = 60 deg

ii) cos(arccos(-sqrt2/2))

Let y = cos(arccos(-sqrt2/2))

arccosy = arccos(-sqrt2/2)

y = -sqrt2/2

iii) y = arctan(tan(3pi/4))

tany = tan(3pi/4)

y = 3pi/4