What is the general solution for sin^2(x)-4sin^2(x)+7cos^2(x)=0 ?

The general solution for sin^2(x)-4sin^2(x)+7cos^2(x)=0 is x=-0.99115…+pin,x=0.99115…+pin

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Popular >

sin^2(x)-4sin^2(x)+7cos^2(x)=0

Solution

sin^{2} (x) - 4 sin^{2} (x) + 7 cos^{2} (x) = 0

Solution

x = - 0.99115 \dots + πn, x = 0.99115 \dots + πn

Degrees

x = - 56.78908 \dots^{\circ} + 18 0^{\circ} n, x = 56.78908 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (x) - 4 sin^{2} (x) + 7 cos^{2} (x) = 0

Factor

sin^{2} (x) - 4 sin^{2} (x) + 7 cos^{2} (x) : (7 cos (x) + 3 sin (x)) (7 cos (x) - 3 sin (x))

sin^{2} (x) - 4 sin^{2} (x) + 7 cos^{2} (x)

Add similar elements:

sin^{2} (x) - 4 sin^{2} (x) = - 3 sin^{2} (x)

= - 3 sin^{2} (x) + 7 cos^{2} (x)

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

7 = (7)^{2}

= (7)^{2} cos^{2} (x) - 3 sin^{2} (x)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (7)^{2} cos^{2} (x) - (3)^{2} sin^{2} (x)

Apply exponent rule:

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

(7)^{2} cos^{2} (x) = (7 cos (x))^{2}

= (7 cos (x))^{2} - (3)^{2} sin^{2} (x)

Apply exponent rule:

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

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

= (7 cos (x))^{2} - (3 sin (x))^{2}

= (7 cos (x))^{2} - (3 sin (x))^{2}

Apply Difference of Two Squares Formula:

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

(7 cos (x))^{2} - (3 sin (x))^{2} = (7 cos (x) + 3 sin (x)) (7 cos (x) - 3 sin (x))

= (7 cos (x) + 3 sin (x)) (7 cos (x) - 3 sin (x))

(7 cos (x) + 3 sin (x)) (7 cos (x) - 3 sin (x)) = 0

Solving each part separately

7 cos (x) + 3 sin (x) = 0 or 7 cos (x) - 3 sin (x) = 0

7 cos (x) + 3 sin (x) = 0 : x = arctan (- \frac{7}{3}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{7 cos ( x ) + 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

7 + 3 tan (x) = 0

7 + 3 tan (x) = 0

Move

7

to the right side

7 + 3 tan (x) = 0

Subtract

7

from both sides

7 + 3 tan (x) - 7 = 0 - 7

Simplify

3 tan (x) = - 7

3 tan (x) = - 7

Divide both sides by

3

3 tan (x) = - 7

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{- 7}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{- 7}{3}

Simplify

\frac{3 tan ( x )}{3} : tan (x)

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 7}{3} : - \frac{7}{3}

\frac{- 7}{3}

Apply the fraction rule:

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

= - \frac{7}{3}

Combine same powers :

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

= - \frac{7}{3}

tan (x) = - \frac{7}{3}

tan (x) = - \frac{7}{3}

tan (x) = - \frac{7}{3}

Apply trig inverse properties

tan (x) = - \frac{7}{3}

General solutions for

tan (x) = - \frac{7}{3}

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

x = arctan (- \frac{7}{3}) + πn

x = arctan (- \frac{7}{3}) + πn

7 cos (x) - 3 sin (x) = 0 : x = arctan (\frac{7}{3}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{7 cos ( x ) - 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

7 - 3 tan (x) = 0

7 - 3 tan (x) = 0

Move

7

to the right side

7 - 3 tan (x) = 0

Subtract

7

from both sides

7 - 3 tan (x) - 7 = 0 - 7

Simplify

- 3 tan (x) = - 7

- 3 tan (x) = - 7

Divide both sides by

- 3

- 3 tan (x) = - 7

Divide both sides by

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 7}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} = \frac{- 7}{- 3}

Simplify

\frac{- 3 tan ( x )}{- 3} : tan (x)

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

Apply the fraction rule:

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 7}{- 3} : \frac{7}{3}

\frac{- 7}{- 3}

Apply the fraction rule:

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

= \frac{7}{3}

Combine same powers :

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

= \frac{7}{3}

tan (x) = \frac{7}{3}

tan (x) = \frac{7}{3}

tan (x) = \frac{7}{3}

Apply trig inverse properties

tan (x) = \frac{7}{3}

General solutions for

tan (x) = \frac{7}{3}

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

x = arctan (\frac{7}{3}) + πn

x = arctan (\frac{7}{3}) + πn

Combine all the solutions

x = arctan (- \frac{7}{3}) + πn, x = arctan (\frac{7}{3}) + πn

Show solutions in decimal form

x = - 0.99115 \dots + πn, x = 0.99115 \dots + πn

Popular Examples

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)-4sin^2(x)+7cos^2(x)=0 ?
The general solution for sin^2(x)-4sin^2(x)+7cos^2(x)=0 is x=-0.99115…+pin,x=0.99115…+pin