English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

4sin^2(x)-4cos^2(x)=2sqrt(3)

Solution

4 sin^{2} (x) - 4 cos^{2} (x) = 23

Solution

x = 1.30899 \dots + 2 πn, x = π - 1.30899 \dots + 2 πn, x = - 1.30899 \dots + 2 πn, x = π + 1.30899 \dots + 2 πn

Degrees

x = 7 5^{\circ} + 36 0^{\circ} n, x = 10 5^{\circ} + 36 0^{\circ} n, x = - 7 5^{\circ} + 36 0^{\circ} n, x = 25 5^{\circ} + 36 0^{\circ} n

Solution steps

4 sin^{2} (x) - 4 cos^{2} (x) = 23

Subtract

23

from both sides

4 sin^{2} (x) - 4 cos^{2} (x) - 23 = 0

Rewrite using trig identities

- 23 - 4 cos^{2} (x) + 4 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 23 - 4 (1 - sin^{2} (x)) + 4 sin^{2} (x)

Simplify

- 23 - 4 (1 - sin^{2} (x)) + 4 sin^{2} (x) : - 23 - 4 + 8 sin^{2} (x)

- 23 - 4 (1 - sin^{2} (x)) + 4 sin^{2} (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

4 \cdot 1 = 4

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

= - 23 - 4 + 4 sin^{2} (x) + 4 sin^{2} (x)

Add similar elements:

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

= - 23 - 4 + 8 sin^{2} (x)

= - 23 - 4 + 8 sin^{2} (x)

- 4 - 23 + 8 sin^{2} (x) = 0

Solve by substitution

- 4 - 23 + 8 sin^{2} (x) = 0

Let:

sin (x) = u

- 4 - 23 + 8 u^{2} = 0

- 4 - 23 + 8 u^{2} = 0 : u = \frac{2 + 3}{2}, u = - \frac{2 + 3}{2}

- 4 - 23 + 8 u^{2} = 0

Move

4

to the right side

- 4 - 23 + 8 u^{2} = 0

Add

4

to both sides

- 4 - 23 + 8 u^{2} + 4 = 0 + 4

Simplify

- 23 + 8 u^{2} = 4

- 23 + 8 u^{2} = 4

Move

23

to the right side

- 23 + 8 u^{2} = 4

Add

23

to both sides

- 23 + 8 u^{2} + 23 = 4 + 23

Simplify

8 u^{2} = 4 + 23

8 u^{2} = 4 + 23

Divide both sides by

8

8 u^{2} = 4 + 23

Divide both sides by

8

\frac{8 u ^{2}}{8} = \frac{4}{8} + \frac{2 3}{8}

Simplify

\frac{8 u ^{2}}{8} = \frac{4}{8} + \frac{2 3}{8}

Simplify

\frac{8 u ^{2}}{8} : u^{2}

\frac{8 u ^{2}}{8}

Divide the numbers:

\frac{8}{8} = 1

= u^{2}

Simplify

\frac{4}{8} + \frac{2 3}{8} : \frac{2 + 3}{4}

\frac{4}{8} + \frac{2 3}{8}

Apply rule

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

= \frac{4 + 2 3}{8}

Factor

4 + 23 : 2 (2 + 3)

4 + 23

Rewrite as

= 2 \cdot 2 + 23

Factor out common term

2

= 2 (2 + 3)

= \frac{2 ( 2 + 3 )}{8}

Cancel the common factor:

2

= \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:

n \frac{a}{b} = \frac{n a}{n b},

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:

n a^{n} = a

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:

n \frac{a}{b} = \frac{n a}{n b},

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:

n a^{n} = a

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 = 1.30899 \dots + 2 πn, x = π - 1.30899 \dots + 2 πn, x = - 1.30899 \dots + 2 πn, x = π + 1.30899 \dots + 2 πn

Graph

Popular Examples

sec^2(θ)=1

sec^{2} (θ) = 1

cos((x+pi)/2)=sin(x)

cos (\frac{x + π}{2}) = sin (x)

2sin(2x)-1=0,0<= x<2pi

2 sin (2 x) - 1 = 0, 0 \leq x < 2 π

cos(A)=-4/5

cos (A) = - \frac{4}{5}

4cot^2(x)=cos^2(x)+sin^2(x)-1

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