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

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

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

Solution

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

Solution

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

Degrees

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

Solution steps

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

Factor

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

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

Break the expression into groups

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

Check

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

TrueCheck

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

False

u = 1, v = - 2

Group into

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

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

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

Factor out

sin (x)

from

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

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

Factor out

- cos (x)

from

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

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

Factor out common term

- cos (x)

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

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

Factor out common term

2 sin (x) + cos (x)

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

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

Solving each part separately

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

Move

1

to the right side

2 tan (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

2 tan (x) = - 1

2 tan (x) = - 1

Divide both sides by

2

2 tan (x) = - 1

Divide both sides by

2

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

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

Combine all the solutions

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

Show solutions in decimal form

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

Popular Examples

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

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

2sqrt(2)sin(x)+sec(x)=0

22 sin (x) + sec (x) = 0

cos(θ)=(-2)/((sqrt(30)+(7)))

cos (θ) = \frac{- 2}{( 30 + ( 7 ) )}

cos^3(x)-2sin(x)-0.7=0

cos^{3} (x) - 2 sin (x) - 0.7 = 0

cos^2(x)-sin^2(x)=sec^2(x)-csc^2(x)

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

Frequently Asked Questions (FAQ)

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