What is the general solution for sqrt(3)cos(x)-sin(x)=sqrt(2) ?

The general solution for sqrt(3)cos(x)-sin(x)=sqrt(2) is x=-1.30899…+2pin,x=0.26179…+2pin

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sqrt(3)cos(x)-sin(x)=sqrt(2)

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

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

Solution steps

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

Add

sin (x)

to both sides

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

Square both sides

(3 cos (x))^{2} = (2 + sin (x))^{2}

Subtract

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

from both sides

3 cos^{2} (x) - 2 - 22 sin (x) - sin^{2} (x) = 0

Rewrite using trig identities

1 - 4 sin^{2} (x) - 2 sin (x) 2 = 0

Solve by substitution

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

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

Show solutions in decimal form

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

cos (x) = \frac{34}{70}

x, z = cos (x^{2} + 2 y)

- 3 sin (t) + 4 cos (t) = 0

cos (θ) = \frac{8}{6}

tan (x) = \frac{- 2 2}{2}

What is the general solution for sqrt(3)cos(x)-sin(x)=sqrt(2) ?
The general solution for sqrt(3)cos(x)-sin(x)=sqrt(2) is x=-1.30899…+2pin,x=0.26179…+2pin