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Popular Trigonometry >

3cos(x)csc(x)=2sqrt(3)cos(x)

Solution

3 cos (x) csc (x) = 23 cos (x)

Solution

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

+1

Degrees

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

Solution steps

3 cos (x) csc (x) = 23 cos (x)

Subtract

23 cos (x)

from both sides

3 cos (x) csc (x) - 23 cos (x) = 0

Factor

3 cos (x) csc (x) - 23 cos (x) : cos (x) (3 csc (x) - 23)

3 cos (x) csc (x) - 23 cos (x)

Factor out common term

cos (x)

= cos (x) (3 csc (x) - 23)

cos (x) (3 csc (x) - 23) = 0

Solving each part separately

cos (x) = 0 or 3 csc (x) - 23 = 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 csc (x) - 23 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

3 csc (x) - 23 = 0

Move

23

to the right side

3 csc (x) - 23 = 0

Add

23

to both sides

3 csc (x) - 23 + 23 = 0 + 23

Simplify

3 csc (x) = 23

3 csc (x) = 23

Divide both sides by

3

3 csc (x) = 23

Divide both sides by

3

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

Simplify

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

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

General solutions for

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

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

Combine all the solutions

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

Graph

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