Question

Solve each equation where 0</=x<2(pie). State exact values.

tan^2 x + tanx = 0

sin2x +  sinx = 0

3cos^2 x - sin^2 x = 2

3csc^2 x + 5cscx - 2 = 0

Answer 1

Tan^2x+Tanx = 0

Take common out Tanx.

Tanx(Tanx+1) = 0

Tanx = 0 and Tanx+1 = 0

Tanx = Tan 0 and Tanx = -1

x = 0 and Tanx = -Tan 45

x = n∏ and x = -n∏/4

Where n belongs z.

Answer 2

3Cos^2x -Sin^2x = 2

We know that Sin^2x = 1-Cos^2x

3Cos^2x-(1-Cos^2x) = 2

3Cos^2x-1+cos^2x = 2

4Cos^2x-1 = 2

Add 1 to each side.

4Cos^2x-1+1 = 2+1

4Cos^2x = 3

Divide to each side by 4.

Cos^2x = 3/4

Apply squreroot on each side.

Cosx = √3/2

Cosx = Cos30

x = 30

Answer 3

  

The trigonometric equation sin2x +  sinx = 0

2 sinx cosx + sinx = 0

sinx [ 2 cosx + 1] = 0

sinx = 0 and 2 cosx + 1 = 0

For sinx = 0

The genaral solution of sin(x) = 0 is x = n₁π , where n₁ is an integer.

For 2 cosx + 1 = 0

2 cosx = -1

cosx = -1/2

The function cosx is negitive second and third quadrant.

In second quadrant (π/2 < x < π)
- 1/2 = - cos(π/3) = cos(π - π/3) = cos(2π/3).

The genaral solution of cos(x) = cos(α) is x = 2n₂π ± α, where n₂ is an integer.
If α = 2π/3, then x = 2n₂π ± 2π/3.

The general solutions of sin2x + sinx = 0 are x = n₁π  and x = 2n₂π + 2π/3 where n₁, n₂∈ Z.