What is the general solution for 5sin(y)-5=3sin(y)-4 ?

The general solution for 5sin(y)-5=3sin(y)-4 is y= pi/6+2pin,y=(5pi)/6+2pin

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Popular Trigonometry >

5sin(y)-5=3sin(y)-4

Solution

5 sin (y) - 5 = 3 sin (y) - 4

Solution

y = \frac{π}{6} + 2 πn, y = \frac{5 π}{6} + 2 πn

Degrees

y = 3 0^{\circ} + 36 0^{\circ} n, y = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (y) - 5 = 3 sin (y) - 4

Solve by substitution

5 sin (y) - 5 = 3 sin (y) - 4

Let:

sin (y) = u

5 u - 5 = 3 u - 4

5 u - 5 = 3 u - 4 : u = \frac{1}{2}

5 u - 5 = 3 u - 4

Move

5

to the right side

5 u - 5 = 3 u - 4

Add

5

to both sides

5 u - 5 + 5 = 3 u - 4 + 5

Simplify

5 u = 3 u + 1

5 u = 3 u + 1

Move

3 u

to the left side

5 u = 3 u + 1

Subtract

3 u

from both sides

5 u - 3 u = 3 u + 1 - 3 u

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = sin (y)

sin (y) = \frac{1}{2}

sin (y) = \frac{1}{2}

sin (y) = \frac{1}{2} : y = \frac{π}{6} + 2 πn, y = \frac{5 π}{6} + 2 πn

sin (y) = \frac{1}{2}

General solutions for

sin (y) = \frac{1}{2}

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}

y = \frac{π}{6} + 2 πn, y = \frac{5 π}{6} + 2 πn

y = \frac{π}{6} + 2 πn, y = \frac{5 π}{6} + 2 πn

Combine all the solutions

y = \frac{π}{6} + 2 πn, y = \frac{5 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5sin(y)-5=3sin(y)-4 ?
The general solution for 5sin(y)-5=3sin(y)-4 is y= pi/6+2pin,y=(5pi)/6+2pin