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Popular >

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

Solution

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

Solution

x = \frac{π}{4} + \frac{2 πn}{3}, x = \frac{5 π}{12} + \frac{2 πn}{3}

Degrees

x = 4 5^{\circ} + 12 0^{\circ} n, x = 7 5^{\circ} + 12 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= cos (2 x + x)

Simplify

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

= - \frac{2}{2}

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

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

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

General solutions for

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

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

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

Solve

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

2 x + x = \frac{3 π}{4} + 2 πn

Add similar elements:

2 x + x = 3 x

3 x = \frac{3 π}{4} + 2 πn

Divide both sides by

3

3 x = \frac{3 π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}

Solve

2 x + x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{12} + \frac{2 πn}{3}

2 x + x = \frac{5 π}{4} + 2 πn

Add similar elements:

2 x + x = 3 x

3 x = \frac{5 π}{4} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{4} + 2 πn

Divide both sides by

3

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

Simplify

\frac{3 x}{3} = \frac{\frac{5 π}{4}}{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}{4}}{3} + \frac{2 πn}{3} : \frac{5 π}{12} + \frac{2 πn}{3}

\frac{\frac{5 π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{4}}{3} = \frac{5 π}{12}

\frac{\frac{5 π}{4}}{3}

Apply the fraction rule:

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

= \frac{5 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{5 π}{12}

= \frac{5 π}{12} + \frac{2 πn}{3}

x = \frac{5 π}{12} + \frac{2 πn}{3}

x = \frac{5 π}{12} + \frac{2 πn}{3}

x = \frac{5 π}{12} + \frac{2 πn}{3}

x = \frac{π}{4} + \frac{2 πn}{3}, x = \frac{5 π}{12} + \frac{2 πn}{3}

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