What is the general solution for 5sin(y)cos(y)=2 ?

The general solution for 5sin(y)cos(y)=2 is y=(0.92729…)/2+pin,y= pi/2-(0.92729…)/2+pin

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Popular Trigonometry >

5sin(y)cos(y)=2

Solution

5 sin (y) cos (y) = 2

Solution

y = \frac{0.92729 \dots}{2} + πn, y = \frac{π}{2} - \frac{0.92729 \dots}{2} + πn

Degrees

y = 26.56505 \dots^{\circ} + 18 0^{\circ} n, y = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 sin (y) cos (y) = 2

Rewrite using trig identities

5 sin (y) cos (y)

Use the Double Angle identity:

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

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

= 5 \cdot \frac{sin ( 2 y )}{2}

5 \cdot \frac{sin ( 2 y )}{2} = 2

Refine

5 \cdot \frac{sin ( 2 y )}{2} : \frac{5 sin ( 2 y )}{2}

5 \cdot \frac{sin ( 2 y )}{2}

Multiply fractions:

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

= \frac{sin ( 2 y ) \cdot 5}{2}

\frac{5 sin ( 2 y )}{2} = 2

Multiply both sides by

2

\frac{5 sin ( 2 y )}{2} = 2

Multiply both sides by

2

\frac{5 sin ( 2 y )}{2} \cdot 2 = 2 \cdot 2

Simplify

\frac{5 sin ( 2 y )}{2} \cdot 2 = 2 \cdot 2

Simplify

\frac{5 sin ( 2 y )}{2} \cdot 2 : 5 sin (2 y)

\frac{5 sin ( 2 y )}{2} \cdot 2

Multiply fractions:

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

= \frac{5 sin ( 2 y ) \cdot 2}{2}

Cancel the common factor:

2

= 5 sin (2 y)

Simplify

2 \cdot 2 : 4

2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

5 sin (2 y) = 4

5 sin (2 y) = 4

5 sin (2 y) = 4

Divide both sides by

5

5 sin (2 y) = 4

Divide both sides by

5

\frac{5 sin ( 2 y )}{5} = \frac{4}{5}

Simplify

sin (2 y) = \frac{4}{5}

sin (2 y) = \frac{4}{5}

Apply trig inverse properties

sin (2 y) = \frac{4}{5}

General solutions for

sin (2 y) = \frac{4}{5}

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

2 y = arcsin (\frac{4}{5}) + 2 πn, 2 y = π - arcsin (\frac{4}{5}) + 2 πn

2 y = arcsin (\frac{4}{5}) + 2 πn, 2 y = π - arcsin (\frac{4}{5}) + 2 πn

Solve

2 y = arcsin (\frac{4}{5}) + 2 πn : y = \frac{arcsin ( \frac{4}{5} )}{2} + πn

2 y = arcsin (\frac{4}{5}) + 2 πn

Divide both sides by

2

2 y = arcsin (\frac{4}{5}) + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{arcsin ( \frac{4}{5} )}{2} + \frac{2 πn}{2}

Simplify

y = \frac{arcsin ( \frac{4}{5} )}{2} + πn

y = \frac{arcsin ( \frac{4}{5} )}{2} + πn

Solve

2 y = π - arcsin (\frac{4}{5}) + 2 πn : y = \frac{π}{2} - \frac{arcsin ( \frac{4}{5} )}{2} + πn

2 y = π - arcsin (\frac{4}{5}) + 2 πn

Divide both sides by

2

2 y = π - arcsin (\frac{4}{5}) + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{π}{2} - \frac{arcsin ( \frac{4}{5} )}{2} + \frac{2 πn}{2}

Simplify

y = \frac{π}{2} - \frac{arcsin ( \frac{4}{5} )}{2} + πn

y = \frac{π}{2} - \frac{arcsin ( \frac{4}{5} )}{2} + πn

y = \frac{arcsin ( \frac{4}{5} )}{2} + πn, y = \frac{π}{2} - \frac{arcsin ( \frac{4}{5} )}{2} + πn

Show solutions in decimal form

y = \frac{0.92729 \dots}{2} + πn, y = \frac{π}{2} - \frac{0.92729 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5sin(y)cos(y)=2 ?
The general solution for 5sin(y)cos(y)=2 is y=(0.92729…)/2+pin,y= pi/2-(0.92729…)/2+pin