What is the general solution for sin^2(x)-5sin(x)cos(x)+6cos^2(x)=0 ?

The general solution for sin^2(x)-5sin(x)cos(x)+6cos^2(x)=0 is x=1.10714…+pin,x=1.24904…+pin

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Popular >

sin^2(x)-5sin(x)cos(x)+6cos^2(x)=0

Solution

sin^{2} (x) - 5 sin (x) cos (x) + 6 cos^{2} (x) = 0

Solution

x = 1.10714 \dots + πn, x = 1.24904 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (x) - 5 sin (x) cos (x) + 6 cos^{2} (x) = 0

Factor

sin^{2} (x) - 5 sin (x) cos (x) + 6 cos^{2} (x) : (sin (x) - 2 cos (x)) (sin (x) - 3 cos (x))

sin^{2} (x) - 5 sin (x) cos (x) + 6 cos^{2} (x)

Break the expression into groups

sin^{2} (x) - 5 sin (x) cos (x) + 6 cos^{2} (x)

Definition

Factors of

6 : 1, 2, 3, 6

6

Divisors (Factors)

Find the Prime factors of

6 : 2, 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Add the prime factors:

2, 3

Add 1 and the number

6

itself

1, 6

The factors of

6

1, 2, 3, 6

Negative factors of

6 : - 1, - 2, - 3, - 6

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 3, - 6

For every two factors such that

u * v = 6,

check if

u + v = - 5

Check

u = 1, v = 6 : u * v = 6, u + v = 7 \Rightarrow

FalseCheck

u = 2, v = 3 : u * v = 6, u + v = 5 \Rightarrow

False

u = - 2, v = - 3

Group into

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

(sin^{2} (x) - 2 sin (x) cos (x)) + (- 3 sin (x) cos (x) + 6 cos^{2} (x))

= (sin^{2} (x) - 2 sin (x) cos (x)) + (- 3 sin (x) cos (x) + 6 cos^{2} (x))

Factor out

sin (x)

from

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

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

Factor out common term

sin (x)

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

Factor out

- 3 cos (x)

from

- 3 sin (x) cos (x) + 6 cos^{2} (x) : - 3 cos (x) (sin (x) - 2 cos (x))

- 3 sin (x) cos (x) + 6 cos^{2} (x)

Apply exponent rule:

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

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

= - 3 sin (x) cos (x) + 6 cos (x) cos (x)

Rewrite

6

2 \cdot 3

= - 3 sin (x) cos (x) + 2 \cdot 3 cos (x) cos (x)

Factor out common term

- 3 cos (x)

= - 3 cos (x) (sin (x) - 2 cos (x))

= sin (x) (sin (x) - 2 cos (x)) - 3 cos (x) (sin (x) - 2 cos (x))

Factor out common term

sin (x) - 2 cos (x)

= (sin (x) - 2 cos (x)) (sin (x) - 3 cos (x))

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

Solving each part separately

sin (x) - 2 cos (x) = 0 or sin (x) - 3 cos (x) = 0

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

sin (x) - 3 cos (x) = 0 : x = arctan (3) + πn

sin (x) - 3 cos (x) = 0

Rewrite using trig identities

sin (x) - 3 cos (x) = 0

Divide both sides by

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

\frac{sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 3 = 0

Use the basic trigonometric identity:

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

tan (x) - 3 = 0

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

tan (x) - 3 + 3 = 0 + 3

Simplify

tan (x) = 3

tan (x) = 3

Apply trig inverse properties

tan (x) = 3

General solutions for

tan (x) = 3

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

x = arctan (3) + πn

x = arctan (3) + πn

Combine all the solutions

x = arctan (2) + πn, x = arctan (3) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = 1.24904 \dots + πn

Popular Examples

cos(2x)+sqrt(3)cos(x)=2

cos (2 x) + 3 cos (x) = 2

6sin(x)+1=0

6 sin (x) + 1 = 0

pi+3arccos(x+1)=0

π + 3 arccos (x + 1) = 0

5sin(2x)+3cos(2x)=0

5 sin (2 x) + 3 cos (2 x) = 0

solvefor x,sin(2x)-sin(x)=0 solve for

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

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)-5sin(x)cos(x)+6cos^2(x)=0 ?
The general solution for sin^2(x)-5sin(x)cos(x)+6cos^2(x)=0 is x=1.10714…+pin,x=1.24904…+pin