What is the general solution for 3sin(θ)=sqrt(3)sin(θ)+3cos(θ)+3sin(θ) ?

The general solution for 3sin(θ)=sqrt(3)sin(θ)+3cos(θ)+3sin(θ) is θ=(2pi)/3+pin

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Popular >

3sin(θ)=sqrt(3)sin(θ)+3cos(θ)+3sin(θ)

Solution

3 sin (θ) = 3 sin (θ) + 3 cos (θ) + 3 sin (θ)

Solution

θ = \frac{2 π}{3} + πn

Degrees

θ = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (θ) = 3 sin (θ) + 3 cos (θ) + 3 sin (θ)

Subtract

3 sin (θ) + 3 cos (θ) + 3 sin (θ)

from both sides

- 3 sin (θ) - 3 cos (θ) = 0

Rewrite using trig identities

- 3 sin (θ) - 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{- 3 sin ( θ ) - 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

- \frac{3 sin ( θ )}{cos ( θ )} - 3 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 3 tan (θ) - 3 = 0

- 3 tan (θ) - 3 = 0

Move

3

to the right side

- 3 tan (θ) - 3 = 0

Add

3

to both sides

- 3 tan (θ) - 3 + 3 = 0 + 3

Simplify

- 3 tan (θ) = 3

- 3 tan (θ) = 3

Divide both sides by

- 3

- 3 tan (θ) = 3

Divide both sides by

- 3

\frac{- 3 tan ( θ )}{- 3} = \frac{3}{- 3}

Simplify

\frac{- 3 tan ( θ )}{- 3} = \frac{3}{- 3}

Simplify

\frac{- 3 tan ( θ )}{- 3} : tan (θ)

\frac{- 3 tan ( θ )}{- 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{3}{- 3} : - 3

\frac{3}{- 3}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= - 3

tan (θ) = - 3

tan (θ) = - 3

tan (θ) = - 3

General solutions for

tan (θ) = - 3

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{2 π}{3} + πn

θ = \frac{2 π}{3} + πn

Popular Examples

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

2 sin^{2} (x) - 3 sin (x) = 2

tan(x+pi)+2sin(x+pi)=0,0<= x<= 2pi

tan (x + π) + 2 sin (x + π) = 0, 0 \leq x \leq 2 π

sin(a)+sqrt(3)cos(a)=0

sin (a) + 3 cos (a) = 0

2sqrt(2)cos(θ)+3=5

22 cos (θ) + 3 = 5

sin(a)cos(a)=1

sin (a) cos (a) = 1

Frequently Asked Questions (FAQ)

What is the general solution for 3sin(θ)=sqrt(3)sin(θ)+3cos(θ)+3sin(θ) ?
The general solution for 3sin(θ)=sqrt(3)sin(θ)+3cos(θ)+3sin(θ) is θ=(2pi)/3+pin