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Popular >

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

Solution

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

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply FOIL method:

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

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

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

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

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

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

2 \cdot 1 \cdot cos (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (x)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

Use the basic trigonometric identity:

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

= - 1 - \frac{1}{csc ( x )} + 2 cos (x) + 2 cos (x) \frac{1}{csc ( x )}

Simplify

- 1 - \frac{1}{csc ( x )} + 2 cos (x) + 2 cos (x) \frac{1}{csc ( x )} : - 1 + \frac{- 1 + 2 cos ( x )}{csc ( x )} + 2 cos (x)

- 1 - \frac{1}{csc ( x )} + 2 cos (x) + 2 cos (x) \frac{1}{csc ( x )}

2 cos (x) \frac{1}{csc ( x )} = \frac{2 cos ( x )}{csc ( x )}

2 cos (x) \frac{1}{csc ( x )}

Multiply fractions:

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

= \frac{1 \cdot 2 cos ( x )}{csc ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 cos ( x )}{csc ( x )}

= - 1 - \frac{1}{csc ( x )} + 2 cos (x) + \frac{2 cos ( x )}{csc ( x )}

Combine the fractions

- \frac{1}{csc ( x )} + \frac{2 cos ( x )}{csc ( x )} : \frac{- 1 + 2 cos ( x )}{csc ( x )}

Apply rule

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

= \frac{- 1 + 2 cos ( x )}{csc ( x )}

= - 1 + \frac{2 cos ( x ) - 1}{csc ( x )} + 2 cos (x)

= - 1 + \frac{- 1 + 2 cos ( x )}{csc ( x )} + 2 cos (x)

Use the basic trigonometric identity:

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

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

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

Factor

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

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

Rewrite as

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

Factor out common term

(- 1 + 2 cos (x))

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

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

Solving each part separately

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

- 1 + 2 cos (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

- 1 + 2 cos (x) = 0

Move

1

to the right side

- 1 + 2 cos (x) = 0

Add

1

to both sides

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

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

sin (x) = - 1

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

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