Question

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.

Answer 1

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!

Answer 2

`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`