What is the general solution for 2cos(x)+2sin(x)=sqrt(6) ?

The general solution for 2cos(x)+2sin(x)=sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin

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2cos(x)+2sin(x)=sqrt(6)

2 cos (x) + 2 sin (x) = 6

x = 0.26179 \dots + 2 πn, x = 1.30899 \dots + 2 πn

Solution steps

2 cos (x) + 2 sin (x) = 6

Subtract

2 sin (x)

from both sides

2 cos (x) = 6 - 2 sin (x)

Square both sides

(2 cos (x))^{2} = (6 - 2 sin (x))^{2}

Subtract

(6 - 2 sin (x))^{2}

from both sides

4 cos^{2} (x) - 6 + 46 sin (x) - 4 sin^{2} (x) = 0

Rewrite using trig identities

- 2 - 8 sin^{2} (x) + 4 sin (x) 6 = 0

Solve by substitution

sin (x) = \frac{6 - 2}{4}, sin (x) = \frac{6 + 2}{4}

sin (x) = \frac{6 - 2}{4} : x = arcsin (\frac{6 - 2}{4}) + 2 πn, x = π - arcsin (\frac{6 - 2}{4}) + 2 πn

sin (x) = \frac{6 + 2}{4} : x = arcsin (\frac{6 + 2}{4}) + 2 πn, x = π - arcsin (\frac{6 + 2}{4}) + 2 πn

Combine all the solutions

x = arcsin (\frac{6 - 2}{4}) + 2 πn, x = π - arcsin (\frac{6 - 2}{4}) + 2 πn, x = arcsin (\frac{6 + 2}{4}) + 2 πn, x = π - arcsin (\frac{6 + 2}{4}) + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{6 - 2}{4}) + 2 πn, x = arcsin (\frac{6 + 2}{4}) + 2 πn

Show solutions in decimal form

x = 0.26179 \dots + 2 πn, x = 1.30899 \dots + 2 πn

6 tan (θ) + 1 = 0

cot (5 x) = - 1

2 cot (θ) + csc^{2} (θ) = 0

csc (x) cot (x) = \frac{2}{3}

2 + 2 cos (x) = 0

What is the general solution for 2cos(x)+2sin(x)=sqrt(6) ?
The general solution for 2cos(x)+2sin(x)=sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin