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

The general solution for 2sin^2(x)=3cos^2(x) is x=-0.88607…+pin,x=0.88607…+pin

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2sin^2(x)=3cos^2(x)

Solution

2 sin^{2} (x) = 3 cos^{2} (x)

Solution

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

Degrees

x = - 50.76847 \dots^{\circ} + 18 0^{\circ} n, x = 50.76847 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 sin^{2} (x) = 3 cos^{2} (x)

Subtract

3 cos^{2} (x)

from both sides

2 sin^{2} (x) - 3 cos^{2} (x) = 0

Factor

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

2 sin^{2} (x) - 3 cos^{2} (x)

Rewrite

2 sin^{2} (x) - 3 cos^{2} (x)

(2 sin (x))^{2} - (3 cos (x))^{2}

2 sin^{2} (x) - 3 cos^{2} (x)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} sin^{2} (x) - 3 cos^{2} (x)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (2)^{2} sin^{2} (x) - (3)^{2} cos^{2} (x)

Apply exponent rule:

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

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

= (2 sin (x))^{2} - (3)^{2} cos^{2} (x)

Apply exponent rule:

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

(3)^{2} cos^{2} (x) = (3 cos (x))^{2}

= (2 sin (x))^{2} - (3 cos (x))^{2}

= (2 sin (x))^{2} - (3 cos (x))^{2}

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

2 sin (x) + 3 cos (x) = 0 : x = arctan (- \frac{3}{2}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{2 sin ( x ) + 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{2 sin ( x )}{cos ( x )} + 3 = 0

Use the basic trigonometric identity:

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

2 tan (x) + 3 = 0

2 tan (x) + 3 = 0

Move

3

to the right side

2 tan (x) + 3 = 0

Subtract

3

from both sides

2 tan (x) + 3 - 3 = 0 - 3

Simplify

2 tan (x) = - 3

2 tan (x) = - 3

Divide both sides by

2

2 tan (x) = - 3

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{- 3}{2}

Simplify

\frac{2 tan ( x )}{2} = \frac{- 3}{2}

Simplify

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

\frac{- 3}{2} : - \frac{3}{2}

\frac{- 3}{2}

Apply the fraction rule:

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

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

tan (x) = - \frac{3}{2}

tan (x) = - \frac{3}{2}

tan (x) = - \frac{3}{2}

Apply trig inverse properties

tan (x) = - \frac{3}{2}

General solutions for

tan (x) = - \frac{3}{2}

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

2 sin (x) - 3 cos (x) = 0 : x = arctan (\frac{3}{2}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{2 sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{2 sin ( x )}{cos ( x )} - 3 = 0

Use the basic trigonometric identity:

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

2 tan (x) - 3 = 0

2 tan (x) - 3 = 0

Move

3

to the right side

2 tan (x) - 3 = 0

Add

3

to both sides

2 tan (x) - 3 + 3 = 0 + 3

Simplify

2 tan (x) = 3

2 tan (x) = 3

Divide both sides by

2

2 tan (x) = 3

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{3}{2}

Simplify

\frac{2 tan ( x )}{2} = \frac{3}{2}

Simplify

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

\frac{3}{2} : \frac{3}{2}

\frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

tan (x) = \frac{3}{2}

tan (x) = \frac{3}{2}

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

Combine all the solutions

x = arctan (- \frac{3}{2}) + πn, x = arctan (\frac{3}{2}) + πn

Show solutions in decimal form

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)=3cos^2(x) ?
The general solution for 2sin^2(x)=3cos^2(x) is x=-0.88607…+pin,x=0.88607…+pin