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Popular >

2cos^2(x)-2sin^2(x)=sqrt(3)

Solution

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

Solution

x = 0.26179 \dots + 2 πn, x = π - 0.26179 \dots + 2 πn, x = - 0.26179 \dots + 2 πn, x = π + 0.26179 \dots + 2 πn

Degrees

x = 1 5^{\circ} + 36 0^{\circ} n, x = 16 5^{\circ} + 36 0^{\circ} n, x = - 1 5^{\circ} + 36 0^{\circ} n, x = 19 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Add similar elements:

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

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

Move

3

to the right side

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

Add

3

to both sides

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

Simplify

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

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

Divide both sides by

- 4

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

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = - \frac{2}{- 4} + \frac{3}{- 4}

Simplify

\frac{- 4 u ^{2}}{- 4} = - \frac{2}{- 4} + \frac{3}{- 4}

Simplify

\frac{- 4 u ^{2}}{- 4} : u^{2}

\frac{- 4 u ^{2}}{- 4}

Apply the fraction rule:

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

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

Divide the numbers:

\frac{4}{4} = 1

= u^{2}

Simplify

- \frac{2}{- 4} + \frac{3}{- 4} : \frac{2 - 3}{4}

- \frac{2}{- 4} + \frac{3}{- 4}

Apply rule

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

= \frac{- 2 + 3}{- 4}

Apply the fraction rule:

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

- 2 + 3 = - (2 - 3)

= \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{2 - 3}{4}, u = - \frac{2 - 3}{4}

\frac{2 - 3}{4} = \frac{2 - 3}{2}

\frac{2 - 3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 3}{2}

- \frac{2 - 3}{4} = - \frac{2 - 3}{2}

- \frac{2 - 3}{4}

Simplify

\frac{2 - 3}{4} : \frac{2 - 3}{2}

\frac{2 - 3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 3}{2}

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

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

Substitute back

u = sin (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2 - 3}{2}) + 2 πn, x = π - arcsin (\frac{2 - 3}{2}) + 2 πn

x = arcsin (\frac{2 - 3}{2}) + 2 πn, x = π - arcsin (\frac{2 - 3}{2}) + 2 πn

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{2 - 3}{2}) + 2 πn, x = π + arcsin (\frac{2 - 3}{2}) + 2 πn

x = arcsin (- \frac{2 - 3}{2}) + 2 πn, x = π + arcsin (\frac{2 - 3}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{2 - 3}{2}) + 2 πn, x = π - arcsin (\frac{2 - 3}{2}) + 2 πn, x = arcsin (- \frac{2 - 3}{2}) + 2 πn, x = π + arcsin (\frac{2 - 3}{2}) + 2 πn

Show solutions in decimal form

x = 0.26179 \dots + 2 πn, x = π - 0.26179 \dots + 2 πn, x = - 0.26179 \dots + 2 πn, x = π + 0.26179 \dots + 2 πn

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