What is the general solution for cos^2(x)+(sqrt(3))/2 sin(2x)=0 ?

The general solution for cos^2(x)+(sqrt(3))/2 sin(2x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=(5pi)/6+pin

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cos^2(x)+(sqrt(3))/2 sin(2x)=0

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Simplify

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

cos^{2} (x) + \frac{3}{2} sin (2 x)

Multiply

\frac{3}{2} sin (2 x) : \frac{3 sin ( 2 x )}{2}

\frac{3}{2} sin (2 x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

2 cos (x)

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

2 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 = \frac{5 π}{6} + 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

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

Simplify

\frac{3 tan ( x )}{3} : tan (x)

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

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

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

General solutions for

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

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + πn

Graph

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7 cos (2 x) = 7 cos (x)

csc(θ/3)=(2sqrt(3))/3

csc (\frac{θ}{3}) = \frac{2 3}{3}

3tan(θ)-cot(θ)=0

3 tan (θ) - cot (θ) = 0

sin(1/2 x+pi/6)=(sqrt(3))/2

sin (\frac{1}{2} x + \frac{π}{6}) = \frac{3}{2}

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6 sin (3 x) = 2

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)+(sqrt(3))/2 sin(2x)=0 ?
The general solution for cos^2(x)+(sqrt(3))/2 sin(2x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=(5pi)/6+pin