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Popular >

3sin(2y)-4sin(2y)cos(2y)=0

Solution

3 sin (2 y) - 4 sin (2 y) cos (2 y) = 0

Solution

y = πn, y = \frac{π}{2} + πn, y = \frac{0.72273 \dots}{2} + πn, y = π - \frac{0.72273 \dots}{2} + πn

Degrees

y = 0^{\circ} + 18 0^{\circ} n, y = 9 0^{\circ} + 18 0^{\circ} n, y = 20.70481 \dots^{\circ} + 18 0^{\circ} n, y = 159.29518 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (2 y) - 4 sin (2 y) cos (2 y) = 0

Factor

3 sin (2 y) - 4 sin (2 y) cos (2 y) : - sin (2 y) (4 cos (2 y) - 3)

3 sin (2 y) - 4 sin (2 y) cos (2 y)

Factor out common term

- sin (2 y)

= - sin (2 y) (- 3 + 4 cos (2 y))

- sin (2 y) (4 cos (2 y) - 3) = 0

Solving each part separately

sin (2 y) = 0 or 4 cos (2 y) - 3 = 0

sin (2 y) = 0 : y = πn, y = \frac{π}{2} + πn

sin (2 y) = 0

General solutions for

sin (2 y) = 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}

2 y = 0 + 2 πn, 2 y = π + 2 πn

2 y = 0 + 2 πn, 2 y = π + 2 πn

Solve

2 y = 0 + 2 πn : y = πn

2 y = 0 + 2 πn

0 + 2 πn = 2 πn

2 y = 2 πn

Divide both sides by

2

2 y = 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{2 πn}{2}

Simplify

y = πn

y = πn

Solve

2 y = π + 2 πn : y = \frac{π}{2} + πn

2 y = π + 2 πn

Divide both sides by

2

2 y = π + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

y = \frac{π}{2} + πn

y = \frac{π}{2} + πn

y = πn, y = \frac{π}{2} + πn

4 cos (2 y) - 3 = 0 : y = \frac{arccos ( \frac{3}{4} )}{2} + πn, y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

4 cos (2 y) - 3 = 0

Move

3

to the right side

4 cos (2 y) - 3 = 0

Add

3

to both sides

4 cos (2 y) - 3 + 3 = 0 + 3

Simplify

4 cos (2 y) = 3

4 cos (2 y) = 3

Divide both sides by

4

4 cos (2 y) = 3

Divide both sides by

4

\frac{4 cos ( 2 y )}{4} = \frac{3}{4}

Simplify

cos (2 y) = \frac{3}{4}

cos (2 y) = \frac{3}{4}

Apply trig inverse properties

cos (2 y) = \frac{3}{4}

General solutions for

cos (2 y) = \frac{3}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 y = arccos (\frac{3}{4}) + 2 πn, 2 y = 2 π - arccos (\frac{3}{4}) + 2 πn

2 y = arccos (\frac{3}{4}) + 2 πn, 2 y = 2 π - arccos (\frac{3}{4}) + 2 πn

Solve

2 y = arccos (\frac{3}{4}) + 2 πn : y = \frac{arccos ( \frac{3}{4} )}{2} + πn

2 y = arccos (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 y = arccos (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{arccos ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

y = \frac{arccos ( \frac{3}{4} )}{2} + πn

y = \frac{arccos ( \frac{3}{4} )}{2} + πn

Solve

2 y = 2 π - arccos (\frac{3}{4}) + 2 πn : y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

2 y = 2 π - arccos (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 y = 2 π - arccos (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

y = \frac{arccos ( \frac{3}{4} )}{2} + πn, y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

Combine all the solutions

y = πn, y = \frac{π}{2} + πn, y = \frac{arccos ( \frac{3}{4} )}{2} + πn, y = π - \frac{arccos ( \frac{3}{4} )}{2} + πn

Show solutions in decimal form

y = πn, y = \frac{π}{2} + πn, y = \frac{0.72273 \dots}{2} + πn, y = π - \frac{0.72273 \dots}{2} + πn

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