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

The general solution for sin^2(x)=sin(2x) is x=2pin,x=pi+2pin,x=1.10714…+pin

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sin^2(x)=sin(2x)

Solution

sin^{2} (x) = sin (2 x)

Solution

x = 2 πn, x = π + 2 πn, x = 1.10714 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (x) = sin (2 x)

Subtract

sin (2 x)

from both sides

sin^{2} (x) - sin (2 x) = 0

Rewrite using trig identities

- sin (2 x) + sin^{2} (x)

Use the Double Angle identity:

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

= - 2 sin (x) cos (x) + sin^{2} (x)

sin^{2} (x) - 2 cos (x) sin (x) = 0

Factor

sin^{2} (x) - 2 cos (x) sin (x) : sin (x) (sin (x) - 2 cos (x))

sin^{2} (x) - 2 cos (x) sin (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

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

Factor out common term

sin (x)

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

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

Solving each part separately

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

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 2 = 0

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

tan (x) - 2 + 2 = 0 + 2

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (2) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.10714 \dots + πn

Popular Examples

sin(5x+50)=cos(4x-23)

sin (5 x + 50) = cos (4 x - 23)

tan(x)=1.2

tan (x) = 1.2

tan(x)=0.9

tan (x) = 0.9

solvefor x,sin(x)+cos(x)=0 solve for

x, sin (x) + cos (x) = 0

6sin(x)=3

6 sin (x) = 3

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)=sin(2x) ?
The general solution for sin^2(x)=sin(2x) is x=2pin,x=pi+2pin,x=1.10714…+pin