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

The general solution for cos(2x)=2sin^2(x) is x=(5pi)/6+pin,x= pi/6+pin

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

Solution

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

Solution

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

Degrees

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

2 sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

= cos^{2} (x) - 3 sin^{2} (x)

cos^{2} (x) - 3 sin^{2} (x) = 0

Factor

cos^{2} (x) - 3 sin^{2} (x) : (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

cos^{2} (x) - 3 sin^{2} (x)

Rewrite

cos^{2} (x) - 3 sin^{2} (x)

cos^{2} (x) - (3 sin (x))^{2}

cos^{2} (x) - 3 sin^{2} (x)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= cos^{2} (x) - (3)^{2} sin^{2} (x)

Apply exponent rule:

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

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

= cos^{2} (x) - (3 sin (x))^{2}

= cos^{2} (x) - (3 sin (x))^{2}

Apply Difference of Two Squares Formula:

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

cos^{2} (x) - (3 sin (x))^{2} = (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

= (cos (x) + 3 sin (x)) (cos (x) - 3 sin (x))

(cos (x) + 3 sin (x)) (cos (x) - 3 sin (x)) = 0

Solving each part separately

cos (x) + 3 sin (x) = 0 or cos (x) - 3 sin (x) = 0

cos (x) + 3 sin (x) = 0 : x = \frac{5 π}{6} + πn

cos (x) + 3 sin (x) = 0

Rewrite using trig identities

cos (x) + 3 sin (x) = 0

Divide both sides by

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

\frac{cos ( x ) + 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 + \frac{3 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

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

1 + 3 tan (x) = 0

1 + 3 tan (x) = 0

Move

1

to the right side

1 + 3 tan (x) = 0

Subtract

1

from both sides

1 + 3 tan (x) - 1 = 0 - 1

Simplify

3 tan (x) = - 1

3 tan (x) = - 1

Divide both sides by

3

3 tan (x) = - 1

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

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

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

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

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

General solutions for

tan (x) = - \frac{3}{3}

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}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

cos (x) - 3 sin (x) = 0 : x = \frac{π}{6} + πn

cos (x) - 3 sin (x) = 0

Rewrite using trig identities

cos (x) - 3 sin (x) = 0

Divide both sides by

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

\frac{cos ( x ) - 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

1 - 3 tan (x) = 0

1 - 3 tan (x) = 0

Move

1

to the right side

1 - 3 tan (x) = 0

Subtract

1

from both sides

1 - 3 tan (x) - 1 = 0 - 1

Simplify

- 3 tan (x) = - 1

- 3 tan (x) = - 1

Divide both sides by

- 3

- 3 tan (x) = - 1

Divide both sides by

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} : tan (x)

\frac{- 3 tan ( x )}{- 3}

Apply the fraction rule:

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

= \frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

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

\frac{- 1}{- 3}

Apply the fraction rule:

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

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

General solutions for

tan (x) = \frac{3}{3}

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}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

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

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3 tan (x) = 2 cos (x)

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)=2sin^2(x) ?
The general solution for cos(2x)=2sin^2(x) is x=(5pi)/6+pin,x= pi/6+pin