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Popular >

2sqrt(3)cos(θ)sin(θ)-cos(θ)=2cos(θ)

Solution

23 cos (θ) sin (θ) - cos (θ) = 2 cos (θ)

Solution

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

+1

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

23 cos (θ) sin (θ) - cos (θ) = 2 cos (θ)

Subtract

2 cos (θ)

from both sides

23 cos (θ) sin (θ) - 3 cos (θ) = 0

Factor

23 cos (θ) sin (θ) - 3 cos (θ) : cos (θ) (23 sin (θ) - 3)

23 cos (θ) sin (θ) - 3 cos (θ)

Factor out common term

cos (θ)

= cos (θ) (23 sin (θ) - 3)

cos (θ) (23 sin (θ) - 3) = 0

Solving each part separately

cos (θ) = 0 or 23 sin (θ) - 3 = 0

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

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

23 sin (θ) - 3 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{2 π}{3} + 2 πn

23 sin (θ) - 3 = 0

Move

3

to the right side

23 sin (θ) - 3 = 0

Add

3

to both sides

23 sin (θ) - 3 + 3 = 0 + 3

Simplify

23 sin (θ) = 3

23 sin (θ) = 3

Divide both sides by

23

23 sin (θ) = 3

Divide both sides by

23

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

Simplify

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

Simplify

\frac{2 3 sin ( θ )}{2 3} : sin (θ)

\frac{2 3 sin ( θ )}{2 3}

Divide the numbers:

\frac{2}{2} = 1

= \frac{3 sin ( θ )}{3}

Cancel the common factor:

3

= sin (θ)

Simplify

\frac{3}{2 3} : \frac{3}{2}

\frac{3}{2 3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{2 \cdot 3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= \frac{3 ^{1 - \frac{1}{2}}}{2}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{3 ^{\frac{1}{2}}}{2}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{3}{2}

sin (θ) = \frac{3}{2}

sin (θ) = \frac{3}{2}

sin (θ) = \frac{3}{2}

General solutions for

sin (θ) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Combine all the solutions

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

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