What is the general solution for sin(2x)csc(x)=1,0<= x<= 2pi ?

Q: What is the general solution for sin(2x)csc(x)=1,0<= x<= 2pi ?

The general solution for sin(2x)csc(x)=1,0<= x<= 2pi is x= pi/3 ,x=(5pi)/3

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sin(2x)csc(x)=1,0<= x<= 2pi

Solution

sin (2 x) csc (x) = 1, 0 \leq x \leq 2 π

Solution

x = \frac{π}{3}, x = \frac{5 π}{3}

Degrees

x = 6 0^{\circ}, x = 30 0^{\circ}

Solution steps

sin (2 x) csc (x) = 1, 0 \leq x \leq 2 π

Subtract

1

from both sides

sin (2 x) csc (x) - 1 = 0

Rewrite using trig identities

- 1 + csc (x) sin (2 x)

Use the Double Angle identity:

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

= - 1 + csc (x) \cdot 2 sin (x) cos (x)

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= - 1 + 2 cos (x) csc (x) \frac{1}{csc ( x )}

2 cos (x) csc (x) \frac{1}{csc ( x )} = 2 cos (x)

2 cos (x) csc (x) \frac{1}{csc ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2 cos ( x ) csc ( x )}{csc ( x )}

Cancel the common factor:

csc (x)

= 1 \cdot 2 cos (x)

Multiply the numbers:

1 \cdot 2 = 2

= 2 cos (x)

= - 1 + 2 cos (x)

- 1 + 2 cos (x) = 0

Move

1

to the right side

- 1 + 2 cos (x) = 0

Add

1

to both sides

- 1 + 2 cos (x) + 1 = 0 + 1

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

Simplify

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{3}, x = \frac{5 π}{3}

Popular Examples

tan(b)=sqrt(3)

tan (b) = 3

sin(x)= 12/16

sin (x) = \frac{12}{16}

1/(tan(x))-2tan(x)=-1/4

\frac{1}{tan ( x )} - 2 tan (x) = - \frac{1}{4}

tan(a)= 3/2

tan (a) = \frac{3}{2}

cos(x+pi/6)=-1

cos (x + \frac{π}{6}) = - 1

Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)csc(x)=1,0<= x<= 2pi ?
The general solution for sin(2x)csc(x)=1,0<= x<= 2pi is x= pi/3 ,x=(5pi)/3