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Popular >

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

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

sin (x) (2 - 3) : 2 sin (x) - 3 sin (x)

sin (x) (2 - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = sin (x), b = 2, c = 3

= sin (x) \cdot 2 - sin (x) 3

= 2 sin (x) - 3 sin (x)

= 2 - 3 + 2 sin (x) - 3 sin (x) - 2 (1 - 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)

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

2 - 2 = 0

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} + (2 - 3) u - 3 = 0

Solve with the quadratic formula

2 u^{2} + (2 - 3) u - 3 = 0

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

= (2 - 3)^{2} + 4 \cdot 23

Multiply the numbers:

4 \cdot 2 = 8

= (2 - 3)^{2} + 83

Expand

(2 - 3)^{2} + 83 : 7 + 43

(2 - 3)^{2} + 83

(2 - 3)^{2} : 7 - 43

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = 3

= 2^{2} - 2 \cdot 23 + (3)^{2}

Simplify

2^{2} - 2 \cdot 23 + (3)^{2} : 7 - 43

2^{2} - 2 \cdot 23 + (3)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 23 = 43

2 \cdot 23

Multiply the numbers:

2 \cdot 2 = 4

= 43

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 4 - 43 + 3

Add the numbers:

4 + 3 = 7

= 7 - 43

= 7 - 43

= 7 - 43 + 83

Add similar elements:

- 43 + 83 = 43

= 7 + 43

= 7 + 43

= 3 + 43 + 4

= (3)^{2} + 43 + (4)^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= (3)^{2} + 43 + 2^{2}

23 \cdot 2 = 43

23 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 43

= (3)^{2} + 23 \cdot 2 + 2^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(3)^{2} + 23 \cdot 2 + 2^{2} = (3 + 2)^{2}

= (3 + 2)^{2}

Apply radical rule:

(3 + 2)^{2} = 3 + 2

= 3 + 2

u_{1, 2} = \frac{- ( 2 - 3 ) \pm ( 3 + 2 )}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( 2 - 3 ) + 3 + 2}{2 \cdot 2}, u_{2} = \frac{- ( 2 - 3 ) - ( 3 + 2 )}{2 \cdot 2}

u = \frac{- ( 2 - 3 ) + 3 + 2}{2 \cdot 2} : \frac{3}{2}

\frac{- ( 2 - 3 ) + 3 + 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

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

Expand

- (2 - 3) + 3 + 2 : 23

- (2 - 3) + 3 + 2

- (2 - 3) : - 2 + 3

- (2 - 3)

Distribute parentheses

= - (2) - (- 3)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 3

= - 2 + 3 + 3 + 2

Simplify

- 2 + 3 + 3 + 2 : 23

- 2 + 3 + 3 + 2

Add similar elements:

3 + 3 = 23

= - 2 + 23 + 2

- 2 + 2 = 0

= 23

= 23

= \frac{2 3}{4}

Cancel the common factor:

2

= \frac{3}{2}

u = \frac{- ( 2 - 3 ) - ( 3 + 2 )}{2 \cdot 2} : - 1

\frac{- ( 2 - 3 ) - ( 3 + 2 )}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

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

Expand

- (2 - 3) - (3 + 2) : - 4

- (2 - 3) - (3 + 2)

- (2 - 3) : - 2 + 3

- (2 - 3)

Distribute parentheses

= - (2) - (- 3)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 3

= - 2 + 3 - (3 + 2)

- (3 + 2) : - 3 - 2

- (3 + 2)

Distribute parentheses

= - (3) - (2)

Apply minus-plus rules

+ (- a) = - a

= - 3 - 2

= - 2 + 3 - 3 - 2

Simplify

- 2 + 3 - 3 - 2 : - 4

- 2 + 3 - 3 - 2

Add similar elements:

3 - 3 = 0

= - 2 - 2

Subtract the numbers:

- 2 - 2 = - 4

= - 4

= - 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

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

Substitute back

u = sin (x)

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

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

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

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

General solutions for

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

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

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

sin (x) = - 1

General solutions for

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

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