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

The general solution for sin(x)cos(x)+sin(x)=3cos^2(x)+3cos(x) is x=pi+2pin,x=1.24904…+pin

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Popular >

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

Solution

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

Solution

x = π + 2 πn, x = 1.24904 \dots + πn

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Factor

sin (x) cos (x) + sin (x) - 3 cos^{2} (x) - 3 cos (x) : (cos (x) + 1) (sin (x) - 3 cos (x))

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

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

Factor out

- 3 cos (x)

from

- 3 cos^{2} (x) - 3 cos (x) : - 3 cos (x) (cos (x) + 1)

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

Apply exponent rule:

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

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

= - 3 cos (x) cos (x) - 3 cos (x)

Factor out common term

- 3 cos (x)

= - 3 cos (x) (cos (x) + 1)

Factor out

sin (x)

from

sin (x) cos (x) + sin (x) : sin (x) (cos (x) + 1)

sin (x) cos (x) + sin (x)

Factor out common term

sin (x)

= sin (x) (cos (x) + 1)

= - 3 cos (x) (cos (x) + 1) + sin (x) (cos (x) + 1)

Factor out common term

cos (x) + 1

= (cos (x) + 1) (sin (x) - 3 cos (x))

(cos (x) + 1) (sin (x) - 3 cos (x)) = 0

Solving each part separately

cos (x) + 1 = 0 or sin (x) - 3 cos (x) = 0

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Move

1

to the right side

cos (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

cos (x) = - 1

cos (x) = - 1

General solutions for

cos (x) = - 1

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}

x = π + 2 πn

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

Show solutions in decimal form

x = π + 2 πn, x = 1.24904 \dots + πn

Popular Examples

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

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

4tan(θ)+7=0

4 tan (θ) + 7 = 0

arctan(x)+arctan(3x)+arctan(7x)=180

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

2cos(a)=1

2 cos (a) = 1

cos^2(x)=(1/2)

cos^{2} (x) = (\frac{1}{2})

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)cos(x)+sin(x)=3cos^2(x)+3cos(x) ?
The general solution for sin(x)cos(x)+sin(x)=3cos^2(x)+3cos(x) is x=pi+2pin,x=1.24904…+pin