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Popular >

3sin^2(x)-1=cos^4(x)

Solution

3 sin^{2} (x) - 1 = cos^{4} (x)

Solution

x = 0.72368 \dots + 2 πn, x = 2 π - 0.72368 \dots + 2 πn, x = 2.41790 \dots + 2 πn, x = - 2.41790 \dots + 2 πn

Degrees

x = 41.46431 \dots^{\circ} + 36 0^{\circ} n, x = 318.53568 \dots^{\circ} + 36 0^{\circ} n, x = 138.53568 \dots^{\circ} + 36 0^{\circ} n, x = - 138.53568 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin^{2} (x) - 1 = cos^{4} (x)

Subtract

cos^{4} (x)

from both sides

3 sin^{2} (x) - 1 - cos^{4} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 1 - cos^{4} (x) + 3 (1 - cos^{2} (x))

Simplify

- 1 - cos^{4} (x) + 3 (1 - cos^{2} (x)) : - cos^{4} (x) - 3 cos^{2} (x) + 2

- 1 - cos^{4} (x) + 3 (1 - cos^{2} (x))

Expand

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

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

= - 1 - cos^{4} (x) + 3 - 3 cos^{2} (x)

Simplify

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

- 1 - cos^{4} (x) + 3 - 3 cos^{2} (x)

Group like terms

= - cos^{4} (x) - 3 cos^{2} (x) - 1 + 3

Add/Subtract the numbers:

- 1 + 3 = 2

= - cos^{4} (x) - 3 cos^{2} (x) + 2

= - cos^{4} (x) - 3 cos^{2} (x) + 2

= - cos^{4} (x) - 3 cos^{2} (x) + 2

2 - cos^{4} (x) - 3 cos^{2} (x) = 0

Solve by substitution

2 - cos^{4} (x) - 3 cos^{2} (x) = 0

Let:

cos (x) = u

2 - u^{4} - 3 u^{2} = 0

2 - u^{4} - 3 u^{2} = 0 : u = i \frac{3 + 17}{2}, u = - i \frac{3 + 17}{2}, u = \frac{17 - 3}{2}, u = - \frac{17 - 3}{2}

2 - u^{4} - 3 u^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- u^{4} - 3 u^{2} + 2 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- v^{2} - 3 v + 2 = 0

Solve

- v^{2} - 3 v + 2 = 0 : v = - \frac{3 + 17}{2}, v = \frac{17 - 3}{2}

- v^{2} - 3 v + 2 = 0

Solve with the quadratic formula

- v^{2} - 3 v + 2 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 3, c = 2

v_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

v_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

(- 3)^{2} - 4 (- 1) \cdot 2 = 17

(- 3)^{2} - 4 (- 1) \cdot 2

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 2

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

Add the numbers:

9 + 8 = 17

= 17

v_{1, 2} = \frac{- ( - 3 ) \pm 17}{2 ( - 1 )}

Separate the solutions

v_{1} = \frac{- ( - 3 ) + 17}{2 ( - 1 )}, v_{2} = \frac{- ( - 3 ) - 17}{2 ( - 1 )}

v = \frac{- ( - 3 ) + 17}{2 ( - 1 )} : - \frac{3 + 17}{2}

\frac{- ( - 3 ) + 17}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{3 + 17}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 17}{- 2}

Apply the fraction rule:

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

= - \frac{3 + 17}{2}

v = \frac{- ( - 3 ) - 17}{2 ( - 1 )} : \frac{17 - 3}{2}

\frac{- ( - 3 ) - 17}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{3 - 17}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 17}{- 2}

Apply the fraction rule:

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

3 - 17 = - (17 - 3)

= \frac{17 - 3}{2}

The solutions to the quadratic equation are:

v = - \frac{3 + 17}{2}, v = \frac{17 - 3}{2}

v = - \frac{3 + 17}{2}, v = \frac{17 - 3}{2}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = - \frac{3 + 17}{2} : u = i \frac{3 + 17}{2}, u = - i \frac{3 + 17}{2}

u^{2} = - \frac{3 + 17}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - \frac{3 + 17}{2}, u = - - \frac{3 + 17}{2}

Simplify

- \frac{3 + 17}{2} : i \frac{3 + 17}{2}

- \frac{3 + 17}{2}

Apply radical rule:

- a = - 1 a

- \frac{3 + 17}{2} = - 1 \frac{3 + 17}{2}

= - 1 \frac{3 + 17}{2}

Apply imaginary number rule:

- 1 = i

= i \frac{3 + 17}{2}

Simplify

- - \frac{3 + 17}{2} : - i \frac{3 + 17}{2}

- - \frac{3 + 17}{2}

Simplify

- \frac{3 + 17}{2} : i \frac{3 + 17}{2}

- \frac{3 + 17}{2}

Apply radical rule:

- a = - 1 a

- \frac{3 + 17}{2} = - 1 \frac{3 + 17}{2}

= - 1 \frac{3 + 17}{2}

Apply imaginary number rule:

- 1 = i

= i \frac{3 + 17}{2}

= - i \frac{3 + 17}{2}

u = i \frac{3 + 17}{2}, u = - i \frac{3 + 17}{2}

Solve

u^{2} = \frac{17 - 3}{2} : u = \frac{17 - 3}{2}, u = - \frac{17 - 3}{2}

u^{2} = \frac{17 - 3}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{17 - 3}{2}, u = - \frac{17 - 3}{2}

The solutions are

u = i \frac{3 + 17}{2}, u = - i \frac{3 + 17}{2}, u = \frac{17 - 3}{2}, u = - \frac{17 - 3}{2}

Substitute back

u = cos (x)

cos (x) = i \frac{3 + 17}{2}, cos (x) = - i \frac{3 + 17}{2}, cos (x) = \frac{17 - 3}{2}, cos (x) = - \frac{17 - 3}{2}

cos (x) = i \frac{3 + 17}{2}, cos (x) = - i \frac{3 + 17}{2}, cos (x) = \frac{17 - 3}{2}, cos (x) = - \frac{17 - 3}{2}

cos (x) = i \frac{3 + 17}{2} :

No Solution

cos (x) = i \frac{3 + 17}{2}

No Solution

cos (x) = - i \frac{3 + 17}{2} :

No Solution

cos (x) = - i \frac{3 + 17}{2}

No Solution

cos (x) = \frac{17 - 3}{2} : x = arccos \frac{17 - 3}{2} + 2 πn, x = 2 π - arccos \frac{17 - 3}{2} + 2 πn

cos (x) = \frac{17 - 3}{2}

Apply trig inverse properties

cos (x) = \frac{17 - 3}{2}

General solutions for

cos (x) = \frac{17 - 3}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos \frac{17 - 3}{2} + 2 πn, x = 2 π - arccos \frac{17 - 3}{2} + 2 πn

x = arccos \frac{17 - 3}{2} + 2 πn, x = 2 π - arccos \frac{17 - 3}{2} + 2 πn

cos (x) = - \frac{17 - 3}{2} : x = arccos - \frac{17 - 3}{2} + 2 πn, x = - arccos - \frac{17 - 3}{2} + 2 πn

cos (x) = - \frac{17 - 3}{2}

Apply trig inverse properties

cos (x) = - \frac{17 - 3}{2}

General solutions for

cos (x) = - \frac{17 - 3}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos - \frac{17 - 3}{2} + 2 πn, x = - arccos - \frac{17 - 3}{2} + 2 πn

x = arccos - \frac{17 - 3}{2} + 2 πn, x = - arccos - \frac{17 - 3}{2} + 2 πn

Combine all the solutions

x = arccos \frac{17 - 3}{2} + 2 πn, x = 2 π - arccos \frac{17 - 3}{2} + 2 πn, x = arccos - \frac{17 - 3}{2} + 2 πn, x = - arccos - \frac{17 - 3}{2} + 2 πn

Show solutions in decimal form

x = 0.72368 \dots + 2 πn, x = 2 π - 0.72368 \dots + 2 πn, x = 2.41790 \dots + 2 πn, x = - 2.41790 \dots + 2 πn

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