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Popular >

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

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn

+1

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 9.59406 \dots^{\circ} + 36 0^{\circ} n, x = 170.40593 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (2 x) - 1 \cdot cos (x) = 0

Rewrite using trig identities

- 1 \cdot cos (x) + 3 sin (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Simplify

- 1 \cdot cos (x) + 3 \cdot 2 sin (x) cos (x) : - cos (x) + 6 sin (x) cos (x)

- 1 \cdot cos (x) + 3 \cdot 2 sin (x) cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

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

Multiply the numbers:

3 \cdot 2 = 6

= - cos (x) + 6 sin (x) cos (x)

= - cos (x) + 6 sin (x) cos (x)

- cos (x) + 6 cos (x) sin (x) = 0

Factor

- cos (x) + 6 cos (x) sin (x) : cos (x) (6 sin (x) - 1)

- cos (x) + 6 cos (x) sin (x)

Factor out common term

cos (x)

= cos (x) (- 1 + 6 sin (x))

cos (x) (6 sin (x) - 1) = 0

Solving each part separately

cos (x) = 0 or 6 sin (x) - 1 = 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

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

6 sin (x) - 1 = 0

Move

1

to the right side

6 sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

6 sin (x) = 1

6 sin (x) = 1

Divide both sides by

6

6 sin (x) = 1

Divide both sides by

6

\frac{6 sin ( x )}{6} = \frac{1}{6}

Simplify

sin (x) = \frac{1}{6}

sin (x) = \frac{1}{6}

Apply trig inverse properties

sin (x) = \frac{1}{6}

General solutions for

sin (x) = \frac{1}{6}

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

x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn

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