Find ALL solutions...sin (x + pi/4) = 1/2

Question

Find ALL solutions in the interval [0,2π).

Please advise.

Thank you!

sin (x + π/4) = 1/2

√3 tan ( x - π/6) - 1 = 0

Answer 1

desmos.com/calculator/xwqv0tdkaw

Answer 2

sin(x + π/4) = 1/2

Sin^-1 (1/2) = π/6 = reference angle

Sine is positive in quadrants 1 and 2.

So, x + π/4 = π/6 or π - π/6

x = π/6 - π/4 or 5π/6 - π/4

x = -π/12 or 7π/12

Add 2π to the first one to get a value in [0, 2π)

So, x = 23π/12 or 7π/12.

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√3 tan(x - π/6) = 1

tan(x - π/6) = 1 / √3 Tan^-1 (1 / √3) = π / 6 = Reference angle

Tangent is positive in quadrants 1 and 3.

So, x - π/6 = π/6 or π + π/6

x = π/3 or 4π/3