Question

Use the double angle identities to find the following value.

If cosθ = √10/8 and 3π/2 < θ < 2π, find sin2θ

Answer 1

We can use the double angle identity for sine to find sin2θ:

sin2θ = 2sinθcosθ

To use this formula, we first need to find sinθ. We can use the Pythagorean identity to find sinθ:

sin^2θ + cos^2θ = 1

Substituting cosθ = √10/8, we get:

sin^2θ + (√10/8)^2 = 1
sin^2θ + 10/64 = 1
sin^2θ = 1 - 10/64
sin^2θ = 54/64
sinθ = ±√54/8

Since 3π/2 < θ < 2π, we know that θ is in the fourth quadrant, where sine is negative. Therefore:

sinθ = -√54/8

Substituting sinθ and cosθ into the double angle identity for sine, we get:

sin2θ = 2sinθcosθ
= 2(-√54/8)(√10/8)
= -√540/32
= -3√15/16

Therefore, sin2θ = -3√15/16.

Answer 2

We can use the double angle identity for sine to find sin2θ:

sin2θ = 2sinθcosθ

Since cosθ = √10/8 and 3π/2 < θ < 2π, we know that θ is in the fourth quadrant, where sine is negative. We can use the Pythagorean identity to find the value of sinθ:

sin²θ + cos²θ = 1

sin²θ = 1 - cos²θ

sinθ = -√(1 - cos²θ)

Substituting the value of cosθ, we get:

sinθ = -√(1 - (√10/8)²) = -√(1 - 5/16) = -√(11/16) = -√11/4

Now we can plug in the values of sinθ and cosθ into the double angle identity for sine:

sin2θ = 2sinθcosθ = 2(-√11/4)(√10/8) = -√110/16 = -√(11/4)(10/16) = -√(55/8)

Therefore, sin2θ = -√55/8 when cosθ = √10/8 and 3π/2 < θ < 2π.