What is the general solution for sin(x)-cos(x)= 1/2 ?

The general solution for sin(x)-cos(x)= 1/2 is x=0.36136…+2pin+pi/4 ,x=pi-0.36136…+2pin+pi/4

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sin(x)-cos(x)= 1/2

sin (x) - cos (x) = \frac{1}{2}

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

Solution steps

sin (x) - cos (x) = \frac{1}{2}

Rewrite using trig identities

2 sin (x - \frac{π}{4}) = \frac{1}{2}

Divide both sides by

2

sin (x - \frac{π}{4}) = \frac{2}{4}

Apply trig inverse properties

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

Solve

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn : x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Solve

x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn : x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}, x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Show solutions in decimal form

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

arcsin (3 x - 1) = \frac{1}{2}

2 cos (\frac{x}{2}) - 1 = 0

5 cos^{2} (x) - 6 cos (x) + 1 = 0

1 + sec^{2} (x) = tan^{2} (x)

cos (t) = \frac{1}{2}

What is the general solution for sin(x)-cos(x)= 1/2 ?
The general solution for sin(x)-cos(x)= 1/2 is x=0.36136…+2pin+pi/4 ,x=pi-0.36136…+2pin+pi/4