What is the general solution for 36sin(x)cos(x)=-9sqrt(3) ?

The general solution for 36sin(x)cos(x)=-9sqrt(3) is x=(2pi)/3+pin,x=(5pi)/6+pin

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36sin(x)cos(x)=-9sqrt(3)

Solution

36 sin (x) cos (x) = - 93

Solution

x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

36 sin (x) cos (x) = - 93

Rewrite using trig identities

36 sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

= 36 \cdot \frac{sin ( 2 x )}{2}

36 \cdot \frac{sin ( 2 x )}{2} = - 93

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

\frac{sin ( 2 x ) \cdot 36}{2} = - 93

Divide the numbers:

\frac{36}{2} = 18

18 sin (2 x) = - 93

Divide both sides by

18

18 sin (2 x) = - 93

Divide both sides by

18

\frac{18 sin ( 2 x )}{18} = \frac{- 9 3}{18}

Simplify

\frac{18 sin ( 2 x )}{18} = \frac{- 9 3}{18}

Simplify

\frac{18 sin ( 2 x )}{18} : sin (2 x)

\frac{18 sin ( 2 x )}{18}

Divide the numbers:

\frac{18}{18} = 1

= sin (2 x)

Simplify

\frac{- 9 3}{18} : - \frac{3}{2}

\frac{- 9 3}{18}

Apply the fraction rule:

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

= - \frac{9 3}{18}

Cancel the common factor:

9

= - \frac{3}{2}

sin (2 x) = - \frac{3}{2}

sin (2 x) = - \frac{3}{2}

sin (2 x) = - \frac{3}{2}

General solutions for

sin (2 x) = - \frac{3}{2}

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = \frac{4 π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

2 x = \frac{4 π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

Solve

2 x = \frac{4 π}{3} + 2 πn : x = \frac{2 π}{3} + πn

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2} : \frac{2 π}{3} + πn

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{4 π}{3}}{2} = \frac{2 π}{3}

\frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

Solve

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2} : \frac{5 π}{6} + πn

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{3}}{2} = \frac{5 π}{6}

\frac{\frac{5 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

Popular Examples

sin(2x)=cos(4x)

sin (2 x) = cos (4 x)

6cos^2(θ)+7=-17cos(θ)

6 cos^{2} (θ) + 7 = - 17 cos (θ)

0=-9*sin(2x)

0 = - 9 \cdot sin (2 x)

sec^2(x)=4tan^2(x)

sec^{2} (x) = 4 tan^{2} (x)

5sin(x)=-4

5 sin (x) = - 4

Frequently Asked Questions (FAQ)

What is the general solution for 36sin(x)cos(x)=-9sqrt(3) ?
The general solution for 36sin(x)cos(x)=-9sqrt(3) is x=(2pi)/3+pin,x=(5pi)/6+pin