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

The general solution for sin(x)cos(x)=sin^2(x) is x=2pin,x=pi+2pin,x= pi/4+pin

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

sin^{2} (x)

from both sides

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

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

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

Solving each part separately

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

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

1 - tan (x) - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Popular Examples

sin(x)= 17/8

sin (x) = \frac{17}{8}

sqrt(3)csc(θ)=2

3 csc (θ) = 2

4sin(x)+8=8

4 sin (x) + 8 = 8

2cos^2(t)-8sin^2(t)cos(t)-1=0

2 cos^{2} (t) - 8 sin^{2} (t) cos (t) - 1 = 0

sin(B)= 4/5

sin (B) = \frac{4}{5}

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)cos(x)=sin^2(x) ?
The general solution for sin(x)cos(x)=sin^2(x) is x=2pin,x=pi+2pin,x= pi/4+pin