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Popular >

sin(x)(sin(x)-1)=3cos^2(x)

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 228.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3 cos^{2} (x)

from both sides

sin (x) (sin (x) - 1) - 3 cos^{2} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

sin (x) (- 1 + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = sin (x), b = - 1, c = sin (x)

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

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot sin (x) + sin (x) sin (x)

Simplify

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

- 1 \cdot sin (x) + sin (x) sin (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= sin (x)

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

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

3 \cdot 1 = 3

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

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

- 3 - u + 4 u^{2} = 0 : u = 1, u = - \frac{3}{4}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 4, b = - 1, c = - 3

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

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

(- 1)^{2} - 4 \cdot 4 (- 3) = 7

(- 1)^{2} - 4 \cdot 4 (- 3)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 4 \cdot 3

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 4 \cdot 3 = 48

4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 4}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 4}

u = \frac{- ( - 1 ) + 7}{2 \cdot 4} : 1

\frac{- ( - 1 ) + 7}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{1 + 7}{2 \cdot 4}

Add the numbers:

1 + 7 = 8

= \frac{8}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8}{8}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 1 ) - 7}{2 \cdot 4} : - \frac{3}{4}

\frac{- ( - 1 ) - 7}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{1 - 7}{2 \cdot 4}

Subtract the numbers:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 6}{8}

Apply the fraction rule:

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

= - \frac{6}{8}

Cancel the common factor:

2

= - \frac{3}{4}

The solutions to the quadratic equation are:

u = 1, u = - \frac{3}{4}

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = - \frac{3}{4}

sin (x) = 1, sin (x) = - \frac{3}{4}

sin (x) = 1 : x = \frac{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{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

sin (x) = - \frac{3}{4}

Apply trig inverse properties

sin (x) = - \frac{3}{4}

General solutions for

sin (x) = - \frac{3}{4}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

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