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Popular Trigonometry >

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

Solution

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

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

- \frac{1}{4}

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

1 - 4 (1 - 2 sin^{2} (x)) + 4 sin^{2} (x) : 12 sin^{2} (x) - 3

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

Multiply the numbers:

4 \cdot 1 = 4

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

Multiply the numbers:

4 \cdot 2 = 8

= - 4 + 8 sin^{2} (x)

= - 4 + 8 sin^{2} (x)

= 1 - 4 + 8 sin^{2} (x) + 4 sin^{2} (x)

Simplify

1 - 4 + 8 sin^{2} (x) + 4 sin^{2} (x) : 12 sin^{2} (x) - 3

1 - 4 + 8 sin^{2} (x) + 4 sin^{2} (x)

Add similar elements:

8 sin^{2} (x) + 4 sin^{2} (x) = 12 sin^{2} (x)

= 1 - 4 + 12 sin^{2} (x)

Subtract the numbers:

1 - 4 = - 3

= 12 sin^{2} (x) - 3

= 12 sin^{2} (x) - 3

= 12 sin^{2} (x) - 3

- 3 + 12 sin^{2} (x) = 0

Solve by substitution

- 3 + 12 sin^{2} (x) = 0

Let:

sin (x) = u

- 3 + 12 u^{2} = 0

- 3 + 12 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 3 + 12 u^{2} = 0

Move

3

to the right side

- 3 + 12 u^{2} = 0

Add

3

to both sides

- 3 + 12 u^{2} + 3 = 0 + 3

Simplify

12 u^{2} = 3

12 u^{2} = 3

Divide both sides by

12

12 u^{2} = 3

Divide both sides by

12

\frac{12 u ^{2}}{12} = \frac{3}{12}

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 = sin (x)

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Graph

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