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Popular >

sin(x)sin(2x)-2cos(x)cos(2x)=0

Solution

sin (x) sin (2 x) - 2 cos (x) cos (2 x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 35.26438 \dots^{\circ} + 18 0^{\circ} n, x = 35.26438 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) sin (2 x) - 2 cos (x) cos (2 x) = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

2 sin (x) cos (x) sin (x) = 2 sin^{2} (x) cos (x)

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 2 cos (x) sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 2 cos (x) sin^{2} (x)

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

- 2 cos (2 x) cos (x) + 2 cos (x) sin^{2} (x) = 0

Factor

- 2 cos (2 x) cos (x) + 2 cos (x) sin^{2} (x) : 2 cos (x) (- cos (2 x) + sin^{2} (x))

- 2 cos (2 x) cos (x) + 2 cos (x) sin^{2} (x)

Rewrite as

= - 2 cos (x) cos (2 x) + 2 cos (x) sin^{2} (x)

Factor out common term

2 cos (x)

= 2 cos (x) (- cos (2 x) + sin^{2} (x))

2 cos (x) (- cos (2 x) + sin^{2} (x)) = 0

Solving each part separately

cos (x) = 0 or - cos (2 x) + sin^{2} (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

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

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

- cos (2 x) + sin^{2} (x) = 0 : x = arctan (- \frac{2}{2}) + πn, x = arctan (\frac{2}{2}) + πn

- cos (2 x) + sin^{2} (x) = 0

Rewrite using trig identities

- cos (2 x) + sin^{2} (x)

Use the Double Angle identity:

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

= - (cos^{2} (x) - sin^{2} (x)) + sin^{2} (x)

Simplify

- (cos^{2} (x) - sin^{2} (x)) + sin^{2} (x) : - cos^{2} (x) + 2 sin^{2} (x)

- (cos^{2} (x) - sin^{2} (x)) + sin^{2} (x)

- (cos^{2} (x) - sin^{2} (x)) : - cos^{2} (x) + sin^{2} (x)

- (cos^{2} (x) - sin^{2} (x))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - cos^{2} (x) + sin^{2} (x)

= - cos^{2} (x) + sin^{2} (x) + sin^{2} (x)

Add similar elements:

sin^{2} (x) + sin^{2} (x) = 2 sin^{2} (x)

= - cos^{2} (x) + 2 sin^{2} (x)

= - cos^{2} (x) + 2 sin^{2} (x)

- cos^{2} (x) + 2 sin^{2} (x) = 0

Factor

- cos^{2} (x) + 2 sin^{2} (x) : (2 sin (x) + cos (x)) (2 sin (x) - cos (x))

- cos^{2} (x) + 2 sin^{2} (x)

Rewrite

2 sin^{2} (x) - cos^{2} (x)

(2 sin (x))^{2} - cos^{2} (x)

2 sin^{2} (x) - cos^{2} (x)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 sin (x))^{2} - cos^{2} (x) = (2 sin (x) + cos (x)) (2 sin (x) - cos (x))

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

(2 sin (x) + cos (x)) (2 sin (x) - cos (x)) = 0

Solving each part separately

2 sin (x) + cos (x) = 0 or 2 sin (x) - cos (x) = 0

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{2}{2}) + πn

2 sin (x) + cos (x) = 0

Rewrite using trig identities

2 sin (x) + cos (x) = 0

Divide both sides by

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

\frac{2 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{2 sin ( x )}{cos ( x )} + 1 = 0

Use the basic trigonometric identity:

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

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

Move

1

to the right side

2 tan (x) + 1 = 0

Subtract

1

from both sides

2 tan (x) + 1 - 1 = 0 - 1

Simplify

2 tan (x) = - 1

2 tan (x) = - 1

Divide both sides by

2

2 tan (x) = - 1

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{- 1}{2}

Simplify

\frac{2 tan ( x )}{2} = \frac{- 1}{2}

Simplify

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

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

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

tan (x) = - \frac{2}{2}

tan (x) = - \frac{2}{2}

tan (x) = - \frac{2}{2}

Apply trig inverse properties

tan (x) = - \frac{2}{2}

General solutions for

tan (x) = - \frac{2}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{2}{2}) + πn

x = arctan (- \frac{2}{2}) + πn

2 sin (x) - cos (x) = 0 : x = arctan (\frac{2}{2}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{2 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

2 tan (x) - 1 = 0

2 tan (x) - 1 = 0

Move

1

to the right side

2 tan (x) - 1 = 0

Add

1

to both sides

2 tan (x) - 1 + 1 = 0 + 1

Simplify

2 tan (x) = 1

2 tan (x) = 1

Divide both sides by

2

2 tan (x) = 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

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

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

tan (x) = \frac{2}{2}

tan (x) = \frac{2}{2}

tan (x) = \frac{2}{2}

Apply trig inverse properties

tan (x) = \frac{2}{2}

General solutions for

tan (x) = \frac{2}{2}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{2}{2}) + πn

x = arctan (\frac{2}{2}) + πn

Combine all the solutions

x = arctan (- \frac{2}{2}) + πn, x = arctan (\frac{2}{2}) + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arctan (- \frac{2}{2}) + πn, x = arctan (\frac{2}{2}) + πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

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