What is the general solution for sin(x)(2-sin(x))=cos^2(x) ?

The general solution for sin(x)(2-sin(x))=cos^2(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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

Solution

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

Solution

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

Degrees

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

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Expand

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

sin (x) (2 - sin (x))

Apply the distributive law:

a (b - c) = ab - a c

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

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

= 2 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)

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

= 2 sin (x) - 1

= 2 sin (x) - 1

= 2 sin (x) - 1

- 1 + 2 sin (x) = 0

Move

1

to the right side

- 1 + 2 sin (x) = 0

Add

1

to both sides

- 1 + 2 sin (x) + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

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

Simplify

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

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

Popular Examples

sin^2(θ)= 7/9

sin^{2} (θ) = \frac{7}{9}

1-tan^2(θ)=0

1 - tan^{2} (θ) = 0

sqrt(3)-tan(3θ)=0

3 - tan (3 θ) = 0

3cot(3x)-sqrt(3)=0

3 cot (3 x) - 3 = 0

3+sin(y)=3cos^2(y)

3 + sin (y) = 3 cos^{2} (y)

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)(2-sin(x))=cos^2(x) ?
The general solution for sin(x)(2-sin(x))=cos^2(x) is x= pi/6+2pin,x=(5pi)/6+2pin