What is the general solution for 2sin^2(θ)-3sin(θ)cos(θ)+cos^2(θ)=0 ?

The general solution for 2sin^2(θ)-3sin(θ)cos(θ)+cos^2(θ)=0 is θ=0.46364…+pin,θ= pi/4+pin

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2sin^2(θ)-3sin(θ)cos(θ)+cos^2(θ)=0

Solution

2 sin^{2} (θ) - 3 sin (θ) cos (θ) + cos^{2} (θ) = 0

Solution

θ = 0.46364 \dots + πn, θ = \frac{π}{4} + πn

Degrees

θ = 26.56505 \dots^{\circ} + 18 0^{\circ} n, θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

2 sin^{2} (θ) - 3 sin (θ) cos (θ) + cos^{2} (θ) = 0

Factor

2 sin^{2} (θ) - 3 sin (θ) cos (θ) + cos^{2} (θ) : (2 sin (θ) - cos (θ)) (sin (θ) - cos (θ))

2 sin^{2} (θ) - 3 sin (θ) cos (θ) + cos^{2} (θ)

Break the expression into groups

2 sin^{2} (θ) - 3 sin (θ) cos (θ) + cos^{2} (θ)

Definition

Factors of

2 : 1, 2

2

Divisors (Factors)

Find the Prime factors of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Add 1

1

The factors of

2

1, 2

Negative factors of

2 : - 1, - 2

Multiply the factors by

- 1

to get the negative factors

- 1, - 2

For every two factors such that

u * v = 2,

check if

u + v = - 3

Check

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

FalseCheck

u = - 1, v = - 2 : u * v = 2, u + v = - 3 \Rightarrow

True

u = - 1, v = - 2

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

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

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

Factor out

sin (θ)

from

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

2 sin^{2} (θ) - sin (θ) cos (θ)

Apply exponent rule:

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

sin^{2} (θ) = sin (θ) sin (θ)

= 2 sin (θ) sin (θ) - sin (θ) cos (θ)

Factor out common term

sin (θ)

= sin (θ) (2 sin (θ) - cos (θ))

Factor out

- cos (θ)

from

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

- 2 sin (θ) cos (θ) + cos^{2} (θ)

Apply exponent rule:

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

cos^{2} (θ) = cos (θ) cos (θ)

= - 2 sin (θ) cos (θ) + cos (θ) cos (θ)

Factor out common term

- cos (θ)

= - cos (θ) (2 sin (θ) - cos (θ))

= sin (θ) (2 sin (θ) - cos (θ)) - cos (θ) (2 sin (θ) - cos (θ))

Factor out common term

2 sin (θ) - cos (θ)

= (2 sin (θ) - cos (θ)) (sin (θ) - cos (θ))

(2 sin (θ) - cos (θ)) (sin (θ) - cos (θ)) = 0

Solving each part separately

2 sin (θ) - cos (θ) = 0 or sin (θ) - cos (θ) = 0

2 sin (θ) - cos (θ) = 0 : θ = arctan (\frac{1}{2}) + πn

2 sin (θ) - cos (θ) = 0

Rewrite using trig identities

2 sin (θ) - cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{2 sin ( θ )}{cos ( θ )} - 1 = 0

Use the basic trigonometric identity:

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

2 tan (θ) - 1 = 0

2 tan (θ) - 1 = 0

Move

1

to the right side

2 tan (θ) - 1 = 0

Add

1

to both sides

2 tan (θ) - 1 + 1 = 0 + 1

Simplify

2 tan (θ) = 1

2 tan (θ) = 1

Divide both sides by

2

2 tan (θ) = 1

Divide both sides by

2

\frac{2 tan ( θ )}{2} = \frac{1}{2}

Simplify

tan (θ) = \frac{1}{2}

tan (θ) = \frac{1}{2}

Apply trig inverse properties

tan (θ) = \frac{1}{2}

General solutions for

tan (θ) = \frac{1}{2}

tan (x) = a \Rightarrow x = arctan (a) + πn

θ = arctan (\frac{1}{2}) + πn

θ = arctan (\frac{1}{2}) + πn

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

sin (θ) - cos (θ) = 0

Rewrite using trig identities

sin (θ) - cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{sin ( θ )}{cos ( θ )} - 1 = 0

Use the basic trigonometric identity:

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

tan (θ) - 1 = 0

tan (θ) - 1 = 0

Move

1

to the right side

tan (θ) - 1 = 0

Add

1

to both sides

tan (θ) - 1 + 1 = 0 + 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 = arctan (\frac{1}{2}) + πn, θ = \frac{π}{4} + πn

Show solutions in decimal form

θ = 0.46364 \dots + πn, θ = \frac{π}{4} + πn

Popular Examples

sin^2(x)+4cos^2(x)=1

sin^{2} (x) + 4 cos^{2} (x) = 1

4cos(x^2)-3=0

4 cos (x^{2}) - 3 = 0

sin(x)+1=0,0<= x<2pi

sin (x) + 1 = 0, 0 \leq x < 2 π

solvefor θ,sin^2(θ)= 1/4 solve for

θ, sin^{2} (θ) = \frac{1}{4}

cot(θ)=sqrt(3),0<= θ<2pi

cot (θ) = 3, 0 \leq θ < 2 π

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(θ)-3sin(θ)cos(θ)+cos^2(θ)=0 ?
The general solution for 2sin^2(θ)-3sin(θ)cos(θ)+cos^2(θ)=0 is θ=0.46364…+pin,θ= pi/4+pin