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

The general solution for sin(2x)=5sin(x) is x=2pin,x=pi+2pin

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Popular Trigonometry >

sin(2x)=5sin(x)

Solution

sin (2 x) = 5 sin (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (2 x) = 5 sin (x)

Subtract

5 sin (x)

from both sides

sin (2 x) - 5 sin (x) = 0

Rewrite using trig identities

sin (2 x) - 5 sin (x)

Use the Double Angle identity:

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

= 2 sin (x) cos (x) - 5 sin (x)

- 5 sin (x) + 2 cos (x) sin (x) = 0

Factor

- 5 sin (x) + 2 cos (x) sin (x) : sin (x) (2 cos (x) - 5)

- 5 sin (x) + 2 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 5 + 2 cos (x))

sin (x) (2 cos (x) - 5) = 0

Solving each part separately

sin (x) = 0 or 2 cos (x) - 5 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

2 cos (x) - 5 = 0 :

No Solution

2 cos (x) - 5 = 0

Move

5

to the right side

2 cos (x) - 5 = 0

Add

5

to both sides

2 cos (x) - 5 + 5 = 0 + 5

Simplify

2 cos (x) = 5

2 cos (x) = 5

Divide both sides by

2

2 cos (x) = 5

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{5}{2}

Simplify

cos (x) = \frac{5}{2}

cos (x) = \frac{5}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=5sin(x) ?
The general solution for sin(2x)=5sin(x) is x=2pin,x=pi+2pin