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Popular Trigonometry >

4sin(x)cos(x)-2sqrt(3)*sin(x)-2cos(x)+sqrt(3)=0

Solution

4 sin (x) cos (x) - 23 \cdot sin (x) - 2 cos (x) + 3 = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3 = 0

Factor

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3 : (- 1 + 2 sin (x)) (2 cos (x) - 3)

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3

Factor

4 sin (x) cos (x) - 2 cos (x) : 2 cos (x) (2 sin (x) - 1)

4 sin (x) cos (x) - 2 cos (x)

Rewrite as

= 2 \cdot 2 cos (x) sin (x) - 1 \cdot 2 cos (x)

Factor out common term

2 cos (x)

= 2 cos (x) (2 sin (x) - 1)

Factor

- 23 sin (x) + 3 : 3 (- 2 sin (x) + 1)

- 23 sin (x) + 3

Factor out common term

3

= 3 (- 2 sin (x) + 1)

= 2 cos (x) (2 sin (x) - 1) + 3 (- 2 sin (x) + 1)

Rewrite as

= 2 (- 1 + 2 sin (x)) cos (x) - (- 1 + 2 sin (x)) 3

Factor out common term

(- 1 + 2 sin (x))

= (- 1 + 2 sin (x)) (2 cos (x) - 3)

(- 1 + 2 sin (x)) (2 cos (x) - 3) = 0

Solving each part separately

- 1 + 2 sin (x) = 0 or 2 cos (x) - 3 = 0

- 1 + 2 sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

- 1 + 2 sin (x) = 0

Move

1

to the right side

- 1 + 2 sin (x) = 0

Add

1

to both sides

- 1 + 2 sin (x) + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{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}

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

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

2 cos (x) - 3 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

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

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Graph

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