Find the exact value of the six trigonometric functions for x if #sin(pi/2-x)= -3/5# What are the exact values?

#pi < x < (3pi)/2#

I understand sin= opp/hyp and that since sin is negative the triangle is in the III and IV quadrants.

2 Answers
Nov 17, 2017

The answer is #3.79# radians.

Explanation:

Use the identity

#sin(A - B) = sinAcosB - cosAsinB#.

#sin(pi/2)cosx - cos(pi/2)sinx = -3/5#

#cosx= -3/5#

We want the solutions in quadrant three because the condition #pi <x <3pi/2# must hold true. Let's take

#x= arccos(3/5)#

#x = 36.9˚#

Accordingly, #x = 180 + 36.9 = 216.9˚#.

This can be converted to #216.9˚ xx pi/180 =3.79 radians #

Hopefully this helps!

Nov 17, 2017

#cos(x) = -3/5#
#sec(x) = -5/3#
#sin(x) = -4/5#
#csc(x) = -5/4#
#tan(x) = 4/3#
#cot(x) = 3/4#

Explanation:

Given: #sin(pi/2-x)= -3/5; pi < x < (3pi)/2#

Use the identity #sin(A-B) = sin(A)cos(B)-cos(A)sin(B)# where #A = pi/2 and B= x#:

#sin(pi/2)cos(x)-cos(pi/2)sin(x) = -3/5; pi < x < (3pi)/2#

Use the facts that #sin(pi/2) =1 and cos(pi/2) = 0#:

#cos(x) = -3/5#

Use the identity #sec(x) = 1/cos(x)#:

#sec(x) = -5/3#

Use the identity #sin(x) = +-sqrt(1-cos^2(x))#:

#sin(x) = +-sqrt(1-(-3/5)^2)#

Choose the negative value, because the sine function is negative in the third quadrant:

#sin(x) = -sqrt(25/25-9/25)#

#sin(x) = -4/5#

Use the identity #csc(x) = 1/sin(x)#:

#csc(x) = -5/4#

Use the identity #tan(x) = sin(x)/cos(x)#:

#tan(x) = (-4/5)/(-3/5)#

#tan(x) = 4/3#

Use the identity #cot(x) = 1/tan(x)#

#cot(x) = 3/4#