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Popular Trigonometry >

3sin^4(x)+cos^4(x)=1

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = - 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Apply exponent rule:

a^{b} = a^{2} a^{b - 2}

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

Rewrite using trig identities

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

Simplify

- 1 + cos^{4} (x) + 3 (1 - cos^{2} (x)) (1 - cos^{2} (x)) : 4 cos^{4} (x) - 6 cos^{2} (x) + 2

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

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

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

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

Add the numbers:

1 + 1 = 2

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

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

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

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

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

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

2 \cdot 1 \cdot cos^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos^{2} (x)

(cos^{2} (x))^{2} = cos^{4} (x)

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (x)

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

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

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

Expand

3 (1 - 2 cos^{2} (x) + cos^{4} (x)) : 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

3 \cdot 1 - 3 \cdot 2 cos^{2} (x) + 3 cos^{4} (x) : 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

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

Multiply the numbers:

3 \cdot 1 = 3

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

Multiply the numbers:

3 \cdot 2 = 6

= 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

= 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

= - 1 + cos^{4} (x) + 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

Simplify

- 1 + cos^{4} (x) + 3 - 6 cos^{2} (x) + 3 cos^{4} (x) : 4 cos^{4} (x) - 6 cos^{2} (x) + 2

- 1 + cos^{4} (x) + 3 - 6 cos^{2} (x) + 3 cos^{4} (x)

Group like terms

= cos^{4} (x) - 6 cos^{2} (x) + 3 cos^{4} (x) - 1 + 3

Add similar elements:

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

= 4 cos^{4} (x) - 6 cos^{2} (x) - 1 + 3

Add/Subtract the numbers:

- 1 + 3 = 2

= 4 cos^{4} (x) - 6 cos^{2} (x) + 2

= 4 cos^{4} (x) - 6 cos^{2} (x) + 2

= 4 cos^{4} (x) - 6 cos^{2} (x) + 2

2 + 4 cos^{4} (x) - 6 cos^{2} (x) = 0

Solve by substitution

2 + 4 cos^{4} (x) - 6 cos^{2} (x) = 0

Let:

cos (x) = u

2 + 4 u^{4} - 6 u^{2} = 0

2 + 4 u^{4} - 6 u^{2} = 0 : u = 1, u = - 1, u = \frac{1}{2}, u = - \frac{1}{2}

2 + 4 u^{4} - 6 u^{2} = 0

Write in the standard form

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

4 u^{4} - 6 u^{2} + 2 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

4 v^{2} - 6 v + 2 = 0

Solve

4 v^{2} - 6 v + 2 = 0 : v = 1, v = \frac{1}{2}

4 v^{2} - 6 v + 2 = 0

Solve with the quadratic formula

4 v^{2} - 6 v + 2 = 0

Quadratic Equation Formula:

For

a = 4, b = - 6, c = 2

v_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

v_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

(- 6)^{2} - 4 \cdot 4 \cdot 2 = 2

(- 6)^{2} - 4 \cdot 4 \cdot 2

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

Subtract the numbers:

36 - 32 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

v_{1, 2} = \frac{- ( - 6 ) \pm 2}{2 \cdot 4}

Separate the solutions

v_{1} = \frac{- ( - 6 ) + 2}{2 \cdot 4}, v_{2} = \frac{- ( - 6 ) - 2}{2 \cdot 4}

v = \frac{- ( - 6 ) + 2}{2 \cdot 4} : 1

\frac{- ( - 6 ) + 2}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{6 + 2}{2 \cdot 4}

Add the numbers:

6 + 2 = 8

= \frac{8}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8}{8}

Apply rule

\frac{a}{a} = 1

= 1

v = \frac{- ( - 6 ) - 2}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 6 ) - 2}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{6 - 2}{2 \cdot 4}

Subtract the numbers:

6 - 2 = 4

= \frac{4}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

The solutions to the quadratic equation are:

v = 1, v = \frac{1}{2}

v = 1, v = \frac{1}{2}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Solve

u^{2} = \frac{1}{2} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{2}, u = - \frac{1}{2}

The solutions are

u = 1, u = - 1, u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = cos (x)

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

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

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

General solutions for

cos (x) = - 1

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) = \frac{1}{2} : x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

cos (x) = - \frac{1}{2} : x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn, x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

Graph

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