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

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

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-3sqrt(3)sin(v)+3cos(v)=3sqrt(2)

- 33 sin (v) + 3 cos (v) = 32

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

Solution steps

- 33 sin (v) + 3 cos (v) = 32

Add

33 sin (v)

to both sides

3 cos (v) = 32 + 33 sin (v)

Square both sides

(3 cos (v))^{2} = (32 + 33 sin (v))^{2}

Subtract

(32 + 33 sin (v))^{2}

from both sides

9 cos^{2} (v) - 18 - 186 sin (v) - 27 sin^{2} (v) = 0

Rewrite using trig identities

- 9 - 36 sin^{2} (v) - 18 sin (v) 6 = 0

Solve by substitution

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

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

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

Combine all the solutions

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

Verify solutions by plugging them into the original equation

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

Show solutions in decimal form

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

x, - arctan (y) = arctan (x)

cot (x) = tan^{2} (x)

sin (x + b) = cos (x)

3 sin (ϕ) - cos (2 ϕ) = 1

2 tan^{2} (θ) + 3 tan (θ) + 4 = 3

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