What is the general solution for cos(x)=(2-tan(x))(1+sin(x)) ?

The general solution for cos(x)=(2-tan(x))(1+sin(x)) is x= pi/3+2pin,x=(5pi)/3+2pin

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Popular Trigonometry >

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

Solution

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

Solution

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

Degrees

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

Solution steps

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

Subtract

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

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplify

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

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

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

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

Join

2 - \frac{sin ( x )}{cos ( x )} : \frac{2 cos ( x ) - sin ( x )}{cos ( x )}

2 - \frac{sin ( x )}{cos ( x )}

Convert element to fraction:

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

= \frac{2 cos ( x )}{cos ( x )} - \frac{sin ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 cos ( x ) - sin ( x )}{cos ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Expand

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

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

Distribute parentheses

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

Apply minus-plus rules

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

= 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

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

Factor

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

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

Factor out common term

sin (x)

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

Subtract

1

from 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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)=(2-tan(x))(1+sin(x)) ?
The general solution for cos(x)=(2-tan(x))(1+sin(x)) is x= pi/3+2pin,x=(5pi)/3+2pin