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Popular >

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

Solution

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

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

3 sin (x) cos (x) - 2 cos (x)

Factor out common term

cos (x)

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

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

Solving each part separately

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

3 sin (x) - 2 = 0

Move

2

to the right side

3 sin (x) - 2 = 0

Add

2

to both sides

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

Simplify

3 sin (x) = 2

3 sin (x) = 2

Divide both sides by

3

3 sin (x) = 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

sin (x) = cos (x) + 1

5 sin (x) + 2 = 0

4 - sec^{2} (x) = 0

3 tan^{2} (x) - 9 = 0

3 cos (4 x) = 2