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Popular >

sin(x)cos(x^2)=sin(x),0<= x<= 2pi

Solution

sin (x) cos (x^{2}) = sin (x), 0 \leq x \leq 2 π

Solution

x = 0, x = π, x = 2 π, x = 2 π, x = 2 π, x = 6 π, x = 22 π

+1

Degrees

x = 0^{\circ}, x = 18 0^{\circ}, x = 36 0^{\circ}, x = 143.61922 \dots^{\circ}, x = 203.10825 \dots^{\circ}, x = 248.75578 \dots^{\circ}, x = 287.23844 \dots^{\circ}

Solution steps

sin (x) cos (x^{2}) = sin (x), 0 \leq x \leq 2 π

Subtract

sin (x)

from both sides

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

Factor

sin (x) cos (x^{2}) - sin (x) : sin (x) (cos (x^{2}) - 1)

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

Factor out common term

sin (x)

= sin (x) (cos (x^{2}) - 1)

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

Solving each part separately

sin (x) = 0 or cos (x^{2}) - 1 = 0

sin (x) = 0, 0 \leq x \leq 2 π : x = 0, x = π, x = 2 π

sin (x) = 0, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = π, x = 2 π

cos (x^{2}) - 1 = 0, 0 \leq x \leq 2 π : x = 0, x = 0, x = 2 π, x = 2 π, x = 6 π, x = 22 π

cos (x^{2}) - 1 = 0, 0 \leq x \leq 2 π

Move

1

to the right side

cos (x^{2}) - 1 = 0

Add

1

to both sides

cos (x^{2}) - 1 + 1 = 0 + 1

Simplify

cos (x^{2}) = 1

cos (x^{2}) = 1

General solutions for

cos (x^{2}) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x^{2} = 0 + 2 πn

x^{2} = 0 + 2 πn

Solve

x^{2} = 0 + 2 πn : x = 2 πn, x = - 2 πn

x^{2} = 0 + 2 πn

0 + 2 πn = 2 πn

x^{2} = 2 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = 2 πn, x = - 2 πn

Simplify

2 πn : 2 πn

2 πn

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 2 πn

Simplify

- 2 πn : - 2 πn

- 2 πn

Apply radical rule:

assuming

a \geq 0, b \geq 0

= - 2 πn

x = 2 πn, x = - 2 πn

x = 2 πn, x = - 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = 0, x = 2 π, x = 2 π, x = 6 π, x = 22 π

Combine all the solutions

x = 0, x = π, x = 2 π, x = 2 π, x = 2 π, x = 6 π, x = 22 π

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