What is the general solution for sin(x)+cos(x)= 1/5 ?

The general solution for sin(x)+cos(x)= 1/5 is x=0.14189…+2pin-pi/4 ,x=pi-0.14189…+2pin-pi/4

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sin(x)+cos(x)= 1/5

sin (x) + cos (x) = \frac{1}{5}

x = 0.14189 \dots + 2 πn - \frac{π}{4}, x = π - 0.14189 \dots + 2 πn - \frac{π}{4}

Solution steps

sin (x) + cos (x) = \frac{1}{5}

Rewrite using trig identities

2 sin (x + \frac{π}{4}) = \frac{1}{5}

Divide both sides by

2

sin (x + \frac{π}{4}) = \frac{2}{10}

Apply trig inverse properties

x + \frac{π}{4} = arcsin (\frac{2}{10}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2}{10}) + 2 πn

Solve

x + \frac{π}{4} = arcsin (\frac{2}{10}) + 2 πn : x = arcsin (\frac{1}{5 2}) + 2 πn - \frac{π}{4}

Solve

x + \frac{π}{4} = π - arcsin (\frac{2}{10}) + 2 πn : x = π - arcsin (\frac{1}{5 2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{1}{5 2}) + 2 πn - \frac{π}{4}, x = π - arcsin (\frac{1}{5 2}) + 2 πn - \frac{π}{4}

Show solutions in decimal form

x = 0.14189 \dots + 2 πn - \frac{π}{4}, x = π - 0.14189 \dots + 2 πn - \frac{π}{4}

cos (\frac{2 x}{3}) - 1 = 0

4 sin^{2} (θ) - 11 sin (θ) + 7 = 0

cos^{2} (x) + cos^{4} (x) = 1

- 3 cos (θ) = 2

tan (x) = \frac{1}{3} 3

What is the general solution for sin(x)+cos(x)= 1/5 ?
The general solution for sin(x)+cos(x)= 1/5 is x=0.14189…+2pin-pi/4 ,x=pi-0.14189…+2pin-pi/4