What is the general solution for cos^2(θ)-sin(θ)cos(θ)=0 ?

The general solution for cos^2(θ)-sin(θ)cos(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin,θ= pi/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(θ)-sin(θ)cos(θ)=0

Solution

cos^{2} (θ) - sin (θ) cos (θ) = 0

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{4} + πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

cos^{2} (θ) - sin (θ) cos (θ) = 0

Factor

cos^{2} (θ) - sin (θ) cos (θ) : cos (θ) (cos (θ) - sin (θ))

cos^{2} (θ) - sin (θ) cos (θ)

Apply exponent rule:

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

cos^{2} (θ) = cos (θ) cos (θ)

= cos (θ) cos (θ) - sin (θ) cos (θ)

Factor out common term

cos (θ)

= cos (θ) (cos (θ) - sin (θ))

cos (θ) (cos (θ) - sin (θ)) = 0

Solving each part separately

cos (θ) = 0 or cos (θ) - sin (θ) = 0

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

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

cos (θ) - sin (θ) = 0 : θ = \frac{π}{4} + πn

cos (θ) - sin (θ) = 0

Rewrite using trig identities

cos (θ) - sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{cos ( θ ) - sin ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

1 - \frac{sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

1 - tan (θ) = 0

1 - tan (θ) = 0

Move

1

to the right side

1 - tan (θ) = 0

Subtract

1

from both sides

1 - tan (θ) - 1 = 0 - 1

Simplify

- tan (θ) = - 1

- tan (θ) = - 1

Divide both sides by

- 1

- tan (θ) = - 1

Divide both sides by

- 1

\frac{- tan ( θ )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (θ) = 1

tan (θ) = 1

General solutions for

tan (θ) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{4} + πn

Popular Examples

solvefor x,cos(2x)+cos(x)=0 solve for

x, cos (2 x) + cos (x) = 0

2sin(θ)+cos(2θ)=1

2 sin (θ) + cos (2 θ) = 1

sin(pit)=0

sin (π t) = 0

8sin^3(x)=8sin(x)

8 sin^{3} (x) = 8 sin (x)

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

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

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(θ)-sin(θ)cos(θ)=0 ?
The general solution for cos^2(θ)-sin(θ)cos(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin,θ= pi/4+pin