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Popular >

cos(2x)= 3/4-5cos^2(x)

Solution

cos (2 x) = \frac{3}{4} - 5 cos^{2} (x)

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (2 x) = \frac{3}{4} - 5 cos^{2} (x)

Subtract

\frac{3}{4} - 5 cos^{2} (x)

from both sides

cos (2 x) - \frac{3}{4} + 5 cos^{2} (x) = 0

Simplify

cos (2 x) - \frac{3}{4} + 5 cos^{2} (x) : \frac{4 cos ( 2 x ) - 3 + 20 cos ^{2} ( x )}{4}

cos (2 x) - \frac{3}{4} + 5 cos^{2} (x)

Convert element to fraction:

cos (2 x) = \frac{cos ( 2 x ) 4}{4}, 5 cos^{2} (x) = \frac{5 cos ^{2} ( x ) 4}{4}

= \frac{cos ( 2 x ) \cdot 4}{4} - \frac{3}{4} + \frac{5 cos ^{2} ( x ) \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( 2 x ) \cdot 4 - 3 + 5 cos ^{2} ( x ) \cdot 4}{4}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{4 cos ( 2 x ) - 3 + 20 cos ^{2} ( x )}{4}

\frac{4 cos ( 2 x ) - 3 + 20 cos ^{2} ( x )}{4} = 0

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

4 cos (2 x) - 3 + 20 cos^{2} (x) = 0

Rewrite using trig identities

- 3 + 20 cos^{2} (x) + 4 cos (2 x)

Use the Double Angle identity:

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

= - 3 + 20 cos^{2} (x) + 4 (2 cos^{2} (x) - 1)

Simplify

- 3 + 20 cos^{2} (x) + 4 (2 cos^{2} (x) - 1) : 28 cos^{2} (x) - 7

- 3 + 20 cos^{2} (x) + 4 (2 cos^{2} (x) - 1)

Expand

4 (2 cos^{2} (x) - 1) : 8 cos^{2} (x) - 4

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 2 cos^{2} (x), c = 1

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

Simplify

4 \cdot 2 cos^{2} (x) - 4 \cdot 1 : 8 cos^{2} (x) - 4

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

Multiply the numbers:

4 \cdot 2 = 8

= 8 cos^{2} (x) - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 8 cos^{2} (x) - 4

= 8 cos^{2} (x) - 4

= - 3 + 20 cos^{2} (x) + 8 cos^{2} (x) - 4

Simplify

- 3 + 20 cos^{2} (x) + 8 cos^{2} (x) - 4 : 28 cos^{2} (x) - 7

- 3 + 20 cos^{2} (x) + 8 cos^{2} (x) - 4

Add similar elements:

20 cos^{2} (x) + 8 cos^{2} (x) = 28 cos^{2} (x)

= - 3 + 28 cos^{2} (x) - 4

Group like terms

= 28 cos^{2} (x) - 3 - 4

Subtract the numbers:

- 3 - 4 = - 7

= 28 cos^{2} (x) - 7

= 28 cos^{2} (x) - 7

= 28 cos^{2} (x) - 7

- 7 + 28 cos^{2} (x) = 0

Solve by substitution

- 7 + 28 cos^{2} (x) = 0

Let:

cos (x) = u

- 7 + 28 u^{2} = 0

- 7 + 28 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 7 + 28 u^{2} = 0

Move

7

to the right side

- 7 + 28 u^{2} = 0

Add

7

to both sides

- 7 + 28 u^{2} + 7 = 0 + 7

Simplify

28 u^{2} = 7

28 u^{2} = 7

Divide both sides by

28

28 u^{2} = 7

Divide both sides by

28

\frac{28 u ^{2}}{28} = \frac{7}{28}

Simplify

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

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

Substitute back

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

General solutions for

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

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

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

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

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

General solutions for

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

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

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

Combine all the solutions

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

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