Question

If cos θ = `8/17` and θ lies in the 1st quadrant, then the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ) is ______.

Options

• `23/17((sqrt(3) - 1)/2 + 1/sqrt(2))`

• `23/17((sqrt(3) + 1)/2 + 1/sqrt(2))`

• `23/17((sqrt(3) - 1)/2 - 1/sqrt(2))`

• `23/17((sqrt(3) + 1)/2 - 1/sqrt(2))`

Answer 1

If cos θ = `8/17` and θ lies in the 1st quadrant, then the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ) is `underlinebb(23/17((sqrt(3) - 1)/2 + 1/sqrt(2)))`.

Explanation:

Since, cos θ = `8/17` and `0 < θ < π/2`

`\implies` sin θ = `sqrt(1 - 8^2/17^2) = 15/17`

The value of the given expression

= cos 30° . cos θ – sin 30° sin θ + cos 45° cos θ + sin 45° sin θ + cos 120° cos θ + sin 120° sin θ

= `cos θ (sqrt(3)/2 + 1/sqrt(2) - 1/2) - sin θ (1/2 - 1/sqrt(2) - sqrt(3)/2)`

= `8/17 (sqrt(3)/2 + 1/sqrt(2) - 1/sqrt(2)) + 15/17 (sqrt(3)/2 + 1/sqrt(2) - 1/sqrt(2))`

= `23/17 ((sqrt(3) - 1)/2 + 1/sqrt(2))`