What is the general solution for cos(x)+2sin(x)=2 ?

The general solution for cos(x)+2sin(x)=2 is x=0.64350…+2pin,x= pi/2+2pin

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cos(x)+2sin(x)=2

cos (x) + 2 sin (x) = 2

x = 0.64350 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Solution steps

cos (x) + 2 sin (x) = 2

Subtract

2 sin (x)

from both sides

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

Square both sides

cos^{2} (x) = (2 - 2 sin (x))^{2}

Subtract

(2 - 2 sin (x))^{2}

from both sides

cos^{2} (x) - 4 + 8 sin (x) - 4 sin^{2} (x) = 0

Rewrite using trig identities

- 3 - 5 sin^{2} (x) + 8 sin (x) = 0

Solve by substitution

sin (x) = \frac{3}{5}, sin (x) = 1

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn, x = \frac{π}{2} + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{3}{5}) + 2 πn, x = \frac{π}{2} + 2 πn

Show solutions in decimal form

x = 0.64350 \dots + 2 πn, x = \frac{π}{2} + 2 πn

csc (θ) = - 2

3 cos (2 t) + 4 = 1

sin^{2} (x) = 4 - 2 cos^{2} (x)

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

sin (x) cos (x) = \frac{1}{4}

What is the general solution for cos(x)+2sin(x)=2 ?
The general solution for cos(x)+2sin(x)=2 is x=0.64350…+2pin,x= pi/2+2pin