What is the general solution for sin(α)+cos(α)= 1/2 ?

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

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sin(α)+cos(α)= 1/2

sin (α) + cos (α) = \frac{1}{2}

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

Solution steps

sin (α) + cos (α) = \frac{1}{2}

Rewrite using trig identities

2 sin (α + \frac{π}{4}) = \frac{1}{2}

Divide both sides by

2

sin (α + \frac{π}{4}) = \frac{2}{4}

Apply trig inverse properties

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

Solve

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

Solve

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

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

Show solutions in decimal form

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

2 sin^{2} (θ) - cos (2 θ) = 1

cos (x - \frac{π}{3}) = - 1

5 tan (B) - 11 = 0

2 csc (x) = 2

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

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