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Popular Trigonometry >

solvefor y,(sin(3y))/3 =(cos(2x))/2+1/2

Solution

solvefor

Solution

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

Solution steps

\frac{sin ( 3 y )}{3} = \frac{cos ( 2 x )}{2} + \frac{1}{2}

Multiply both sides by

3

\frac{sin ( 3 y )}{3} = \frac{cos ( 2 x )}{2} + \frac{1}{2}

Multiply both sides by

3

\frac{3 sin ( 3 y )}{3} = 3 \cdot \frac{cos ( 2 x )}{2} + 3 \cdot \frac{1}{2}

Simplify

\frac{3 sin ( 3 y )}{3} = 3 \cdot \frac{cos ( 2 x )}{2} + 3 \cdot \frac{1}{2}

Simplify

\frac{3 sin ( 3 y )}{3} : sin (3 y)

\frac{3 sin ( 3 y )}{3}

Divide the numbers:

\frac{3}{3} = 1

= sin (3 y)

Simplify

3 \cdot \frac{cos ( 2 x )}{2} + 3 \cdot \frac{1}{2} : \frac{3 cos ( 2 x ) + 3}{2}

3 \cdot \frac{cos ( 2 x )}{2} + 3 \cdot \frac{1}{2}

3 \cdot \frac{cos ( 2 x )}{2} = \frac{3 cos ( 2 x )}{2}

3 \cdot \frac{cos ( 2 x )}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( 2 x ) \cdot 3}{2}

3 \cdot \frac{1}{2} = \frac{3}{2}

3 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

= \frac{3 cos ( 2 x )}{2} + \frac{3}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 cos ( 2 x ) + 3}{2}

sin (3 y) = \frac{3 cos ( 2 x ) + 3}{2}

sin (3 y) = \frac{3 cos ( 2 x ) + 3}{2}

sin (3 y) = \frac{3 cos ( 2 x ) + 3}{2}

Apply trig inverse properties

sin (3 y) = \frac{3 cos ( 2 x ) + 3}{2}

General solutions for

sin (3 y) = \frac{3 cos ( 2 x ) + 3}{2}

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

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

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

Solve

3 y = arcsin (\frac{3 cos ( 2 x ) + 3}{2}) + 2 πn : y = \frac{arcsin ( \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

3 y = arcsin (\frac{3 cos ( 2 x ) + 3}{2}) + 2 πn

Divide both sides by

3

3 y = arcsin (\frac{3 cos ( 2 x ) + 3}{2}) + 2 πn

Divide both sides by

3

\frac{3 y}{3} = \frac{arcsin ( \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

Simplify

y = \frac{arcsin ( \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

y = \frac{arcsin ( \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

Solve

3 y = π + arcsin (- \frac{3 cos ( 2 x ) + 3}{2}) + 2 πn : y = \frac{π}{3} + \frac{arcsin ( - \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

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

Divide both sides by

3

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

Divide both sides by

3

\frac{3 y}{3} = \frac{π}{3} + \frac{arcsin ( - \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

Simplify

y = \frac{π}{3} + \frac{arcsin ( - \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

y = \frac{π}{3} + \frac{arcsin ( - \frac{3 c o s ( 2 x ) + 3}{2} )}{3} + \frac{2 πn}{3}

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

Graph

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