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

The general solution for cos^2(x)=3sin(x)cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=0.32175…+pin

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Popular >

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

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.32175 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 18.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3 sin (x) cos (x)

from both sides

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

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 = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) - 3 sin (x) = 0 : x = arctan (\frac{1}{3}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - 3 tan (x) = 0

1 - 3 tan (x) = 0

Move

1

to the right side

1 - 3 tan (x) = 0

Subtract

1

from both sides

1 - 3 tan (x) - 1 = 0 - 1

Simplify

- 3 tan (x) = - 1

- 3 tan (x) = - 1

Divide both sides by

- 3

- 3 tan (x) = - 1

Divide both sides by

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

Simplify

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

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

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arctan (\frac{1}{3}) + πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.32175 \dots + πn

Popular Examples

cos(θ)-3cos(2θ)=0

cos (θ) - 3 cos (2 θ) = 0

sin^2(x)+3sin(x)=4

sin^{2} (x) + 3 sin (x) = 4

sin(y)=(50)/(65.3)

sin (y) = \frac{50}{65.3}

0.26=(1-sin(x))/(1+sin(x))

0.26 = \frac{1 - sin ( x )}{1 + sin ( x )}

sec(x)=-2,0<= x<= 2pi

sec (x) = - 2, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)=3sin(x)cos(x) ?
The general solution for cos^2(x)=3sin(x)cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=0.32175…+pin