What is the general solution for cos(2x)=2sin(x)cos(x) ?

The general solution for cos(2x)=2sin(x)cos(x) is x= pi/8+(pin)/2

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

Solution

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

Solution

x = \frac{π}{8} + \frac{πn}{2}

Degrees

x = 22. 5^{\circ} + 9 0^{\circ} n

Solution steps

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

Subtract

2 sin (x) cos (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Divide both sides by

cos (2 x), cos (2 x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (2 x) = 0

1 - tan (2 x) = 0

Move

1

to the right side

1 - tan (2 x) = 0

Subtract

1

from both sides

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

Simplify

- tan (2 x) = - 1

- tan (2 x) = - 1

Divide both sides by

- 1

- tan (2 x) = - 1

Divide both sides by

- 1

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

Simplify

tan (2 x) = 1

tan (2 x) = 1

General solutions for

tan (2 x) = 1

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}

2 x = \frac{π}{4} + πn

2 x = \frac{π}{4} + πn

Solve

2 x = \frac{π}{4} + πn : x = \frac{π}{8} + \frac{πn}{2}

2 x = \frac{π}{4} + πn

Divide both sides by

2

2 x = \frac{π}{4} + πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{4}}{2} + \frac{πn}{2} : \frac{π}{8} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

Popular Examples

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sin (θ) = \frac{5}{6}

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3 tan^{2} (x) = 3 tan (x)

2sec(x)-4cos(x)=0

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cos(x+1)=sin(x)

cos (x + 1) = sin (x)

2sin^2(θ)=2+cos(2θ)

2 sin^{2} (θ) = 2 + cos (2 θ)

Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)=2sin(x)cos(x) ?
The general solution for cos(2x)=2sin(x)cos(x) is x= pi/8+(pin)/2