Solving Equations Using Trigonometric Identities

[trulyfalse]

Hey PF!

Homework Statement

Find exact solutions for the following equations over the domain 0 ≤ x <2π
2sinx = 3 + 2cscx

Homework Equations

sin²+cos²=1

The Attempt at a Solution

2sinx = 3 + 2cscx
2sinx = 3 +2(1/sinx)
sinx = 3/2 + 1/sinx
sinx - 1/sinx = 3/2
(1-1-cos²x)/sinx = 3/2
-cos²x/sinx = 3/2
cos²x/sinx = -3/2

I am perplexed by this question. Where do I go from here? How do I solve this equation?

 

[SammyS]

Hey PF!

Homework Statement

Find exact solutions for the following equations over the domain 0 ≤ x <2π
2sinx = 3 + 2cscx

Homework Equations

sin²+cos²=1

The Attempt at a Solution

2sinx = 3 + 2cscx
2sinx = 3 +2(1/sinx)
sinx = 3/2 + 1/sinx
sinx - 1/sinx = 3/2
(1-1-cos²x)/sinx = 3/2
-cos²x/sinx = 3/2
cos²x/sinx = -3/2

I am perplexed by this question. Where do I go from here? How do I solve this equation?

No need to change sin²(x) into 1-cos²(x) .

Multiply both sides of sin(x) = 3/2 + 1/sin(x) by sin(x) .

 

[trulyfalse]

Ahhhh... Thank you for elucidating me. I can see now that I have to factor and solve for x. Thank you again for your help!