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Popular Trigonometry >

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

Solution

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

Solution

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

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Expand

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

(- cos (x) + 2 sin (x)) (1 + cos (x))

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - cos (x), b = 2 sin (x), c = 1, d = cos (x)

= (- cos (x)) \cdot 1 + (- cos (x)) cos (x) + 2 sin (x) \cdot 1 + 2 sin (x) cos (x)

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot cos (x) - cos (x) cos (x) + 2 \cdot 1 \cdot sin (x) + 2 sin (x) cos (x)

Simplify

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

- 1 \cdot cos (x) - cos (x) cos (x) + 2 \cdot 1 \cdot sin (x) + 2 sin (x) cos (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= cos (x)

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 \cdot sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (x)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Factor

- 1 - cos (x) + 2 sin (x) + 2 cos (x) sin (x) : (cos (x) + 1) (2 sin (x) - 1)

- 1 - cos (x) + 2 sin (x) + 2 cos (x) sin (x)

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

Factor out

2 sin (x)

from

2 sin (x) cos (x) + 2 sin (x) : 2 sin (x) (cos (x) + 1)

2 sin (x) cos (x) + 2 sin (x)

Factor out common term

2 sin (x)

= 2 sin (x) (cos (x) + 1)

Factor out

- 1

from

- cos (x) - 1 : - (cos (x) + 1)

- cos (x) - 1

Factor out common term

- 1

= - (cos (x) + 1)

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

Factor out common term

cos (x) + 1

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

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

Solving each part separately

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

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Move

1

to the right side

cos (x) + 1 = 0

Subtract

1

from both sides

cos (x) + 1 - 1 = 0 - 1

Simplify

cos (x) = - 1

cos (x) = - 1

General solutions for

cos (x) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

2 sin (x) - 1 + 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

Combine all the solutions

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

Graph

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