Question

Do as directed:

2 cos² θ − 1 = 0, find θ, tan θ, sin 2θ and sin² θ.

Answer 1

Given:

2 cos² θ − 1 = 0; find θ, tan θ, sin 2θ and sin² θ.

Step 1: Solve for cos⁡θ.

2 cos² θ = 1

= cos ² θ = `1/2`

= cos θ = ± `1/sqrt2`

= ± `sqrt2/2`

= cos θ = `sqrt2/2 = θ = 45°`

Step 2: tan θ

tan 45° = 1

Step 3: Find sin⁡2θ

sin 2θ = 2 sinθ cosθ

cosθ = `sqrt2/2`

sin θ = `sqrt2/2` (since θ = 45°)

sin 2θ = 2 ⋅ `sqrt2/2 ⋅ sqrt2/2 = 2 ⋅ 2/4 = 1`

Step 4: Find sin⁡²θ

sin² θ = `(sqrt2/2) = 1/2`