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Popular >

sqrt(3)sin(x)sec(x)=2sin(x)

Solution

3 sin (x) sec (x) = 2 sin (x)

Solution

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

+1

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (x) sec (x) = 2 sin (x)

Subtract

2 sin (x)

from both sides

3 sin (x) sec (x) - 2 sin (x) = 0

Factor

3 sin (x) sec (x) - 2 sin (x) : sin (x) (3 sec (x) - 2)

3 sin (x) sec (x) - 2 sin (x)

Factor out common term

sin (x)

= sin (x) (3 sec (x) - 2)

sin (x) (3 sec (x) - 2) = 0

Solving each part separately

sin (x) = 0 or 3 sec (x) - 2 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

3 sec (x) - 2 = 0

Move

2

to the right side

3 sec (x) - 2 = 0

Add

2

to both sides

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

Simplify

3 sec (x) = 2

3 sec (x) = 2

Divide both sides by

3

3 sec (x) = 2

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

= sec (x)

Simplify

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

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

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

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

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

General solutions for

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

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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