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Popular >

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

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

4 - 7 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= - 4 + 2 sin^{4} (x) + 7 (1 - sin^{2} (x))

Simplify

- 4 + 2 sin^{4} (x) + 7 (1 - sin^{2} (x)) : 2 sin^{4} (x) - 7 sin^{2} (x) + 3

- 4 + 2 sin^{4} (x) + 7 (1 - sin^{2} (x))

Expand

7 (1 - sin^{2} (x)) : 7 - 7 sin^{2} (x)

7 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 1, c = sin^{2} (x)

= 7 \cdot 1 - 7 sin^{2} (x)

Multiply the numbers:

7 \cdot 1 = 7

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

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

Simplify

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

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

Group like terms

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

Add/Subtract the numbers:

- 4 + 7 = 3

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

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

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

3 + 2 sin^{4} (x) - 7 sin^{2} (x) = 0

Solve by substitution

3 + 2 sin^{4} (x) - 7 sin^{2} (x) = 0

Let:

sin (x) = u

3 + 2 u^{4} - 7 u^{2} = 0

3 + 2 u^{4} - 7 u^{2} = 0 : u = 3, u = - 3, u = \frac{1}{2}, u = - \frac{1}{2}

3 + 2 u^{4} - 7 u^{2} = 0

Write in the standard form

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

2 u^{4} - 7 u^{2} + 3 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

2 v^{2} - 7 v + 3 = 0

Solve

2 v^{2} - 7 v + 3 = 0 : v = 3, v = \frac{1}{2}

2 v^{2} - 7 v + 3 = 0

Solve with the quadratic formula

2 v^{2} - 7 v + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = - 7, c = 3

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 3 = 5

(- 7)^{2} - 4 \cdot 2 \cdot 3

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

v_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 2}

Separate the solutions

v_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 2}, v_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 2}

v = \frac{- ( - 7 ) + 5}{2 \cdot 2} : 3

\frac{- ( - 7 ) + 5}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 + 5}{2 \cdot 2}

Add the numbers:

7 + 5 = 12

= \frac{12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

v = \frac{- ( - 7 ) - 5}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 7 ) - 5}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 - 5}{2 \cdot 2}

Subtract the numbers:

7 - 5 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

The solutions to the quadratic equation are:

v = 3, v = \frac{1}{2}

v = 3, v = \frac{1}{2}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 3 : u = 3, u = - 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

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 = 3, u = - 3, u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = 3, sin (x) = - 3, sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = 3, sin (x) = - 3, sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = 3 :

No Solution

sin (x) = 3

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - 3 :

No Solution

sin (x) = - 3

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Popular Examples

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sin (\frac{75 π}{11}) = sin (\frac{x π}{11})

sec^2(θ)= 4/3

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