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

The general solution for (cos(x))/(sin(x))=cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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

Solution

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

Solution

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

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

Factor

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

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

Apply exponent rule:

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

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

= cos (x) - cos (x) cos (x)

Factor out common term

cos (x)

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

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

Solving each part separately

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

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

cos (x) = 0

General solutions for

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

1 - sin (x) cos (x) = 0 :

No Solution

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

Rewrite using trig identities

1 - sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

Popular Examples

sin(2x-pi/4)=0

sin (2 x - \frac{π}{4}) = 0

7sin^2(θ)-4sin(θ)=3

7 sin^{2} (θ) - 4 sin (θ) = 3

tan(θ)-cot(θ)=0

tan (θ) - cot (θ) = 0

csc^3(x)+csc^2(x)= 4/3 csc(x)+4/3

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

6sec^2(x)tan(x)=12tan(x)

6 sec^{2} (x) tan (x) = 12 tan (x)

Frequently Asked Questions (FAQ)

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