What is the general solution for 3sin^2(θ)-cos^2(θ)=0 ?

The general solution for 3sin^2(θ)-cos^2(θ)=0 is θ=(5pi)/6+pin,θ= pi/6+pin

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Popular >

3sin^2(θ)-cos^2(θ)=0

Solution

3 sin^{2} (θ) - cos^{2} (θ) = 0

Solution

θ = \frac{5 π}{6} + πn, θ = \frac{π}{6} + πn

Degrees

θ = 15 0^{\circ} + 18 0^{\circ} n, θ = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sin^{2} (θ) - cos^{2} (θ) = 0

Factor

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

3 sin^{2} (θ) - cos^{2} (θ)

Rewrite

3 sin^{2} (θ) - cos^{2} (θ)

(3 sin (θ))^{2} - cos^{2} (θ)

3 sin^{2} (θ) - cos^{2} (θ)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} sin^{2} (θ) - cos^{2} (θ)

Apply exponent rule:

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

(3)^{2} sin^{2} (θ) = (3 sin (θ))^{2}

= (3 sin (θ))^{2} - cos^{2} (θ)

= (3 sin (θ))^{2} - cos^{2} (θ)

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

3 sin (θ) + cos (θ) = 0 : θ = \frac{5 π}{6} + πn

3 sin (θ) + cos (θ) = 0

Rewrite using trig identities

3 sin (θ) + cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{3 sin ( θ )}{cos ( θ )} + 1 = 0

Use the basic trigonometric identity:

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

3 tan (θ) + 1 = 0

3 tan (θ) + 1 = 0

Move

1

to the right side

3 tan (θ) + 1 = 0

Subtract

1

from both sides

3 tan (θ) + 1 - 1 = 0 - 1

Simplify

3 tan (θ) = - 1

3 tan (θ) = - 1

Divide both sides by

3

3 tan (θ) = - 1

Divide both sides by

3

\frac{3 tan ( θ )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( θ )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( θ )}{3} : tan (θ)

\frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (θ) = - \frac{3}{3}

tan (θ) = - \frac{3}{3}

tan (θ) = - \frac{3}{3}

General solutions for

tan (θ) = - \frac{3}{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{5 π}{6} + πn

θ = \frac{5 π}{6} + πn

3 sin (θ) - cos (θ) = 0 : θ = \frac{π}{6} + πn

3 sin (θ) - cos (θ) = 0

Rewrite using trig identities

3 sin (θ) - cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{3 sin ( θ )}{cos ( θ )} - 1 = 0

Use the basic trigonometric identity:

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

3 tan (θ) - 1 = 0

3 tan (θ) - 1 = 0

Move

1

to the right side

3 tan (θ) - 1 = 0

Add

1

to both sides

3 tan (θ) - 1 + 1 = 0 + 1

Simplify

3 tan (θ) = 1

3 tan (θ) = 1

Divide both sides by

3

3 tan (θ) = 1

Divide both sides by

3

\frac{3 tan ( θ )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( θ )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( θ )}{3} : tan (θ)

\frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (θ) = \frac{3}{3}

tan (θ) = \frac{3}{3}

tan (θ) = \frac{3}{3}

General solutions for

tan (θ) = \frac{3}{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}{6} + πn

θ = \frac{π}{6} + πn

Combine all the solutions

θ = \frac{5 π}{6} + πn, θ = \frac{π}{6} + πn

Popular Examples

2cot(θ)-3=0

2 cot (θ) - 3 = 0

1/2 =cos(θ)

\frac{1}{2} = cos (θ)

(sin(2x))/(1-cos(2x))=1

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

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

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

9sec^2(x)-12=0

9 sec^{2} (x) - 12 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(θ)-cos^2(θ)=0 ?
The general solution for 3sin^2(θ)-cos^2(θ)=0 is θ=(5pi)/6+pin,θ= pi/6+pin