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

The general solution for cos(2x)=sin^2(x) is x=-0.61547…+pin,x=0.61547…+pin

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Popular Trigonometry >

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

Solution

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

Solution

x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

Degrees

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

Solution steps

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

Subtract

sin^{2} (x)

from both sides

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) - 2 sin^{2} (x)

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

Solving each part separately

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 + 2 tan (x) = 0

1 + 2 tan (x) = 0

Move

1

to the right side

1 + 2 tan (x) = 0

Subtract

1

from both sides

1 + 2 tan (x) - 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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - 2 tan (x) = 0

1 - 2 tan (x) = 0

Move

1

to the right side

1 - 2 tan (x) = 0

Subtract

1

from both sides

1 - 2 tan (x) - 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}

Apply the fraction rule:

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

= \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

Combine all the solutions

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

Show solutions in decimal form

x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)=sin^2(x) ?
The general solution for cos(2x)=sin^2(x) is x=-0.61547…+pin,x=0.61547…+pin