Question

Use a double-angle or half-angle identity to find the exact value of each expression?

Answer 1

If 0 < θ < π/2, then 0 < θ/2 < π/4, and for such angles we expect sin(θ/2) to be positive. Also, we know both sin(θ) and cos(θ) will be positive.

Given that tan(θ) = 2, we can find sec(θ) from the Pythagorean identity:

tan²(θ) + 1 = sec²(θ) ==> sec(θ) = √(tan²(θ) + 1)

… ==> cos(θ) = 1/√(tan²(θ) + 1)

… ==> cos(θ) = 1/√5

Now, recall the half-angle identity for sine:

sin²(θ/2) = (1 - cos(θ))/2

==> sin(θ/2) = √[(1 - cos(θ))/2]

==> sin(θ/2) = √[(1 - 1/√5)/2]

==> sin(θ/2) = √[(√5 - 1)/(2√5)]

==> sin(θ/2) = √[(5 - √5)/10]