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

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

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Popular >

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

Solution

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

Solution

x = \frac{2 π}{3} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3 sin (x) + 3 cos (x) + 3 sin (x)

from both sides

- 3 sin (x) - 3 cos (x) = 0

Rewrite using trig identities

- 3 sin (x) - 3 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

- 3 tan (x) - 3 = 0

- 3 tan (x) - 3 = 0

Move

3

to the right side

- 3 tan (x) - 3 = 0

Add

3

to both sides

- 3 tan (x) - 3 + 3 = 0 + 3

Simplify

- 3 tan (x) = 3

- 3 tan (x) = 3

Divide both sides by

- 3

- 3 tan (x) = 3

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 tan ( x )}{- 3} : tan (x)

\frac{- 3 tan ( x )}{- 3}

Apply the fraction rule:

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

= \frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

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 (x) = - 3

tan (x) = - 3

tan (x) = - 3

General solutions for

tan (x) = - 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}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

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

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

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

tan(x)+sec(x)=-1

tan (x) + sec (x) = - 1

2sin^2(θ)-cos(θ)-1=0

2 sin^{2} (θ) - cos (θ) - 1 = 0

Frequently Asked Questions (FAQ)

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