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

Solution

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

Solution

x = \frac{5 π}{18} + \frac{2 πn}{3}, x = \frac{7 π}{18} + \frac{2 πn}{3}

Degrees

x = 5 0^{\circ} + 12 0^{\circ} n, x = 7 0^{\circ} + 12 0^{\circ} n

Solution steps

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

Rewrite using trig identities

sin (x) sin (2 x) - cos (x) cos (2 x)

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (x + 2 x)

- cos (x + 2 x) = \frac{3}{2}

Divide both sides by

- 1

- cos (x + 2 x) = \frac{3}{2}

Divide both sides by

- 1

\frac{- cos ( x + 2 x )}{- 1} = \frac{\frac{3}{2}}{- 1}

Simplify

\frac{- cos ( x + 2 x )}{- 1} = \frac{\frac{3}{2}}{- 1}

Simplify

\frac{- cos ( x + 2 x )}{- 1} : cos (x + 2 x)

\frac{- cos ( x + 2 x )}{- 1}

Apply the fraction rule:

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

= \frac{cos ( x + 2 x )}{1}

Apply rule

\frac{a}{1} = a

= cos (x + 2 x)

Simplify

\frac{\frac{3}{2}}{- 1} : - \frac{3}{2}

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

= - \frac{3}{2}

cos (x + 2 x) = - \frac{3}{2}

cos (x + 2 x) = - \frac{3}{2}

cos (x + 2 x) = - \frac{3}{2}

General solutions for

cos (x + 2 x) = - \frac{3}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x + 2 x = \frac{5 π}{6} + 2 πn, x + 2 x = \frac{7 π}{6} + 2 πn

x + 2 x = \frac{5 π}{6} + 2 πn, x + 2 x = \frac{7 π}{6} + 2 πn

Solve

x + 2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{18} + \frac{2 πn}{3}

x + 2 x = \frac{5 π}{6} + 2 πn

Add similar elements:

x + 2 x = 3 x

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3} : \frac{5 π}{18} + \frac{2 πn}{3}

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

\frac{\frac{5 π}{6}}{3} = \frac{5 π}{18}

\frac{\frac{5 π}{6}}{3}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{5 π}{18}

= \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

Solve

x + 2 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{18} + \frac{2 πn}{3}

x + 2 x = \frac{7 π}{6} + 2 πn

Add similar elements:

x + 2 x = 3 x

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3} : \frac{7 π}{18} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} = \frac{7 π}{18}

\frac{\frac{7 π}{6}}{3}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{7 π}{18}

= \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}, x = \frac{7 π}{18} + \frac{2 πn}{3}

Popular Examples

sec(x)-tan(x)=cos(x)

sec (x) - tan (x) = cos (x)

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

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

(0.9)/(cos(x))+(0.7*sin(x))=1.1

\frac{0.9}{cos ( x )} + (0.7 \cdot sin (x)) = 1.1

-sin^2(θ)+cos^2(θ)=1+sin(θ)

- sin^{2} (θ) + cos^{2} (θ) = 1 + sin (θ)

sin(x/2)=1

sin (\frac{x}{2}) = 1