What is the general solution for (sin^2(x))/4 =(cos^2(x))/2 ?

The general solution for (sin^2(x))/4 =(cos^2(x))/2 is x=-0.95531…+pin,x=0.95531…+pin

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(sin^2(x))/4 =(cos^2(x))/2

Solution

\frac{sin ^{2} ( x )}{4} = \frac{cos ^{2} ( x )}{2}

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

\frac{sin ^{2} ( x )}{4} = \frac{cos ^{2} ( x )}{2}

Subtract

\frac{cos ^{2} ( x )}{2}

from both sides

\frac{sin ^{2} ( x )}{4} - \frac{cos ^{2} ( x )}{2} = 0

Simplify

\frac{sin ^{2} ( x )}{4} - \frac{cos ^{2} ( x )}{2} : \frac{sin ^{2} ( x ) - 2 cos ^{2} ( x )}{4}

\frac{sin ^{2} ( x )}{4} - \frac{cos ^{2} ( x )}{2}

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{cos ^{2} ( x )}{2} :

multiply the denominator and numerator by

2

\frac{cos ^{2} ( x )}{2} = \frac{cos ^{2} ( x ) \cdot 2}{2 \cdot 2} = \frac{cos ^{2} ( x ) \cdot 2}{4}

= \frac{sin ^{2} ( x )}{4} - \frac{cos ^{2} ( x ) \cdot 2}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ^{2} ( x ) - cos ^{2} ( x ) \cdot 2}{4}

\frac{sin ^{2} ( x ) - 2 cos ^{2} ( x )}{4} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

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Frequently Asked Questions (FAQ)

What is the general solution for (sin^2(x))/4 =(cos^2(x))/2 ?
The general solution for (sin^2(x))/4 =(cos^2(x))/2 is x=-0.95531…+pin,x=0.95531…+pin