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

The general solution for 2*cos^2(x)-sin^2(x)=0 is x=-0.95531…+pin,x=0.95531…+pin

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Popular >

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

Solution

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

Solution

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

Degrees

x = - 54.73561 \dots^{\circ} + 18 0^{\circ} n, x = 54.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

2 cos (x) + sin (x) = 0 : x = arctan (- 2) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

2 + tan (x) = 0

2 + tan (x) = 0

Move

2

to the right side

2 + tan (x) = 0

Subtract

2

from both sides

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

Simplify

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

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

x = arctan (- 2) + πn

x = arctan (- 2) + πn

2 cos (x) - sin (x) = 0 : x = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

2 - tan (x) = 0

2 - tan (x) = 0

Move

2

to the right side

2 - tan (x) = 0

Subtract

2

from both sides

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

Simplify

- tan (x) = - 2

- tan (x) = - 2

Divide both sides by

- 1

- tan (x) = - 2

Divide both sides by

- 1

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

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = arctan (- 2) + πn, x = arctan (2) + πn

Show solutions in decimal form

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

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cos (θ) = - \frac{35}{6}

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

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2 sin (2 x) = 0

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Frequently Asked Questions (FAQ)

What is the general solution for 2*cos^2(x)-sin^2(x)=0 ?
The general solution for 2*cos^2(x)-sin^2(x)=0 is x=-0.95531…+pin,x=0.95531…+pin