What is the general solution for sin(2x)csc(x)=1 ?

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

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sin(2x)csc(x)=1

sin (2 x) csc (x) = 1

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (2 x) csc (x) = 1

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

tan (θ) = 21

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

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

32 cos (θ) + 1 = - 2

5 sin (2 θ) = 9 tan (θ)

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