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

The general solution for sin(2x)=3cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=0.98279…+pin

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Popular >

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

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.98279 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 56.30993 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

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

- 3 cos (x) + 2 sin (x) = 0 : x = arctan (\frac{3}{2}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

- 3 + 2 tan (x) = 0

- 3 + 2 tan (x) = 0

Move

3

to the right side

- 3 + 2 tan (x) = 0

Add

3

to both sides

- 3 + 2 tan (x) + 3 = 0 + 3

Simplify

2 tan (x) = 3

2 tan (x) = 3

Divide both sides by

2

2 tan (x) = 3

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{3}{2}

Simplify

tan (x) = \frac{3}{2}

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

Combine all the solutions

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

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.98279 \dots + πn

Popular Examples

1/(cos(x))=1.98

\frac{1}{cos ( x )} = 1.98

2sin^2(3θ)-1=0

2 sin^{2} (3 θ) - 1 = 0

4sin(4θ-pi/3)+3=5

4 sin (4 θ - \frac{π}{3}) + 3 = 5

2tan(x)-3=0

2 tan (x) - 3 = 0

-2cos(2x-pi/3)=2sin(2x-pi/6)

- 2 cos (2 x - \frac{π}{3}) = 2 sin (2 x - \frac{π}{6})

Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=3cos^2(x) ?
The general solution for sin(2x)=3cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=0.98279…+pin