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

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

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Popular >

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

Solution

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

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

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

Subtract

2 cos^{2} (x)

from both sides

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

Solving each part separately

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) + 2 = 0

tan (x) + 2 = 0

Move

2

to the right side

tan (x) + 2 = 0

Subtract

2

from both sides

tan (x) + 2 - 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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

tan (x) - 2 + 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

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

Popular Examples

9sin(2θ)-2sin(θ)=0

9 sin (2 θ) - 2 sin (θ) = 0

3csc(θ)-4=0

3 csc (θ) - 4 = 0

cos(2x)-sin(3x)=0

cos (2 x) - sin (3 x) = 0

8sin(θ)-8=4sqrt(3)-8

8 sin (θ) - 8 = 43 - 8

sec(θ)=2cos(θ)

sec (θ) = 2 cos (θ)

Frequently Asked Questions (FAQ)

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