What is the general solution for cos(2θ)-sin^2(θ)=0 ?

The general solution for cos(2θ)-sin^2(θ)=0 is θ=-0.61547…+pin,θ=0.61547…+pin

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Popular >

cos(2θ)-sin^2(θ)=0

Solution

cos (2 θ) - sin^{2} (θ) = 0

Solution

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

Degrees

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

Solution steps

cos (2 θ) - sin^{2} (θ) = 0

Rewrite using trig identities

cos (2 θ) - sin^{2} (θ)

Use the Double Angle identity:

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

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

Simplify

= cos^{2} (θ) - 2 sin^{2} (θ)

cos^{2} (θ) - 2 sin^{2} (θ) = 0

Factor

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

cos^{2} (θ) - 2 sin^{2} (θ)

Rewrite

cos^{2} (θ) - 2 sin^{2} (θ)

cos^{2} (θ) - (2 sin (θ))^{2}

cos^{2} (θ) - 2 sin^{2} (θ)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= cos^{2} (θ) - (2)^{2} sin^{2} (θ)

Apply exponent rule:

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

(2)^{2} sin^{2} (θ) = (2 sin (θ))^{2}

= cos^{2} (θ) - (2 sin (θ))^{2}

= cos^{2} (θ) - (2 sin (θ))^{2}

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

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

cos (θ) + 2 sin (θ) = 0

Rewrite using trig identities

cos (θ) + 2 sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

1 + \frac{2 sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

1 + 2 tan (θ) = 0

1 + 2 tan (θ) = 0

Move

1

to the right side

1 + 2 tan (θ) = 0

Subtract

1

from both sides

1 + 2 tan (θ) - 1 = 0 - 1

Simplify

2 tan (θ) = - 1

2 tan (θ) = - 1

Divide both sides by

2

2 tan (θ) = - 1

Divide both sides by

2

\frac{2 tan ( θ )}{2} = \frac{- 1}{2}

Simplify

\frac{2 tan ( θ )}{2} = \frac{- 1}{2}

Simplify

\frac{2 tan ( θ )}{2} : tan (θ)

\frac{2 tan ( θ )}{2}

Cancel the common factor:

2

= tan (θ)

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 (θ) = - \frac{2}{2}

tan (θ) = - \frac{2}{2}

tan (θ) = - \frac{2}{2}

Apply trig inverse properties

tan (θ) = - \frac{2}{2}

General solutions for

tan (θ) = - \frac{2}{2}

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

θ = arctan (- \frac{2}{2}) + πn

θ = arctan (- \frac{2}{2}) + πn

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

cos (θ) - 2 sin (θ) = 0

Rewrite using trig identities

cos (θ) - 2 sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

1 - \frac{2 sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

1 - 2 tan (θ) = 0

1 - 2 tan (θ) = 0

Move

1

to the right side

1 - 2 tan (θ) = 0

Subtract

1

from both sides

1 - 2 tan (θ) - 1 = 0 - 1

Simplify

- 2 tan (θ) = - 1

- 2 tan (θ) = - 1

Divide both sides by

- 2

- 2 tan (θ) = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 tan ( θ )}{- 2} : tan (θ)

\frac{- 2 tan ( θ )}{- 2}

Apply the fraction rule:

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

= \frac{2 tan ( θ )}{2}

Cancel the common factor:

2

= tan (θ)

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 (θ) = \frac{2}{2}

tan (θ) = \frac{2}{2}

tan (θ) = \frac{2}{2}

Apply trig inverse properties

tan (θ) = \frac{2}{2}

General solutions for

tan (θ) = \frac{2}{2}

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

θ = arctan (\frac{2}{2}) + πn

θ = arctan (\frac{2}{2}) + πn

Combine all the solutions

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

Show solutions in decimal form

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2θ)-sin^2(θ)=0 ?
The general solution for cos(2θ)-sin^2(θ)=0 is θ=-0.61547…+pin,θ=0.61547…+pin