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

The general solution for sin^2(x)-3sin(x)cos(x)+2cos^2(x)=0 is x= pi/4+pin,x=1.10714…+pin

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sin^2(x)-3sin(x)cos(x)+2cos^2(x)=0

Solution

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

Solution

x = \frac{π}{4} + πn, x = 1.10714 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

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

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

Break the expression into groups

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

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})

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

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

Factor out

sin (x)

from

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

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

Factor out

- 2 cos (x)

from

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

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

Apply exponent rule:

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

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

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

Factor out common term

- 2 cos (x)

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

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

Factor out common term

sin (x) - cos (x)

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

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

Solving each part separately

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

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

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 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}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

tan (x) - 2 + 2 = 0 + 2

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (2) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.10714 \dots + πn

Popular Examples

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

sin(t)=-(sqrt(2))/2

sin (t) = - \frac{2}{2}

9cos(2θ)=9cos^2(θ)-4

9 cos (2 θ) = 9 cos^{2} (θ) - 4

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- 10 sin (x) = 0

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sin (\frac{x}{5}) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)-3sin(x)cos(x)+2cos^2(x)=0 ?
The general solution for sin^2(x)-3sin(x)cos(x)+2cos^2(x)=0 is x= pi/4+pin,x=1.10714…+pin