Radian solution

Question

Find all radian solutions to the following equation.

sin( A + pi/12) = - rad2/2

Answer 1

sin( A + π/12) = - (√2)/ 2

(Next time, don't use rad for the symbol of square root, especially if you are also using the word radian in the problem. You can use sqrt())

If the sine of an angle is negative, then it is either located on 3rd or 4th quadrant

Let θ = A + π/12

sin θ = opposite / hypothenuse = y / r = - (√2)/ 2

r² = x² + y²

2² = x² + (- (√2))²

4 = x² + 2

x = √2

Then its a 45° - 45° - 90° triangle. But it's located in 3rd or 4th quadrant. Since π/4 radian = 45°, therefore:

θ = π + π/4 = 5π/4

θ = 2π - π/4 = 7π/4

Put back A + π/12 as a value for θ.

Split it into two equations:

(1)A + π/12 = 5π/4

A = 15π/12 - π/12 = 14π/12 = 7π/6

(2)A + π/12 = 7π/4

A = 21π/12 - π/12 = 20π/12 = 5π/3

Therefore, all radian solutions for A are:

A₁ = 7π/6 + 2π n

A₂ = 5π/3 + 2π n

Where n ∈ ℤ