What is the general solution for sin^2(θ)=3cos^2(θ) ?

The general solution for sin^2(θ)=3cos^2(θ) is θ=(2pi)/3+pin,θ= pi/3+pin

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Popular >

sin^2(θ)=3cos^2(θ)

Solution

sin^{2} (θ) = 3 cos^{2} (θ)

Solution

θ = \frac{2 π}{3} + πn, θ = \frac{π}{3} + πn

Degrees

θ = 12 0^{\circ} + 18 0^{\circ} n, θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (θ) = 3 cos^{2} (θ)

Subtract

3 cos^{2} (θ)

from both sides

sin^{2} (θ) - 3 cos^{2} (θ) = 0

Factor

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

sin^{2} (θ) - 3 cos^{2} (θ)

Rewrite

sin^{2} (θ) - 3 cos^{2} (θ)

sin^{2} (θ) - (3 cos (θ))^{2}

sin^{2} (θ) - 3 cos^{2} (θ)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= sin^{2} (θ) - (3)^{2} cos^{2} (θ)

Apply exponent rule:

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

(3)^{2} cos^{2} (θ) = (3 cos (θ))^{2}

= sin^{2} (θ) - (3 cos (θ))^{2}

= sin^{2} (θ) - (3 cos (θ))^{2}

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

sin (θ) + 3 cos (θ) = 0 : θ = \frac{2 π}{3} + πn

sin (θ) + 3 cos (θ) = 0

Rewrite using trig identities

sin (θ) + 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{sin ( θ )}{cos ( θ )} + 3 = 0

Use the basic trigonometric identity:

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

tan (θ) + 3 = 0

tan (θ) + 3 = 0

Move

3

to the right side

tan (θ) + 3 = 0

Subtract

3

from both sides

tan (θ) + 3 - 3 = 0 - 3

Simplify

tan (θ) = - 3

tan (θ) = - 3

General solutions for

tan (θ) = - 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}

θ = \frac{2 π}{3} + πn

θ = \frac{2 π}{3} + πn

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

sin (θ) - 3 cos (θ) = 0

Rewrite using trig identities

sin (θ) - 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{sin ( θ )}{cos ( θ )} - 3 = 0

Use the basic trigonometric identity:

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

tan (θ) - 3 = 0

tan (θ) - 3 = 0

Move

3

to the right side

tan (θ) - 3 = 0

Add

3

to both sides

tan (θ) - 3 + 3 = 0 + 3

Simplify

tan (θ) = 3

tan (θ) = 3

General solutions for

tan (θ) = 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}

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

Combine all the solutions

θ = \frac{2 π}{3} + πn, θ = \frac{π}{3} + πn

Popular Examples

20sin^2(x)+27cos(x)=6

20 sin^{2} (x) + 27 cos (x) = 6

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

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

arctan(x)=(2.73)/(7.5)

arctan (x) = \frac{2.73}{7.5}

solvefor y,sin(x)+cos(y)=ln(x)solve for

y, sin (x) + cos (y) = ln (x)

-8csc^2(x)-4cot(x)=-12

- 8 csc^{2} (x) - 4 cot (x) = - 12

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(θ)=3cos^2(θ) ?
The general solution for sin^2(θ)=3cos^2(θ) is θ=(2pi)/3+pin,θ= pi/3+pin