What is the general solution for sin(x)cot(x)=0.5 ?

The general solution for sin(x)cot(x)=0.5 is x= pi/3+2pin,x=(5pi)/3+2pin

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sin(x)cot(x)=0.5

sin (x) cot (x) = 0.5

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 (x) cot (x) = 0.5

Subtract

0.5

from both sides

sin (x) cot (x) - 0.5 = 0

Rewrite using trig identities

- 0.5 + cot (x) sin (x)

Use the basic trigonometric identity:

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

= - 0.5 + \frac{cos ( x )}{sin ( x )} sin (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= cos (x)

= - 0.5 + cos (x)

- 0.5 + cos (x) = 0

Move

0.5

to the right side

- 0.5 + cos (x) = 0

Add

0.5

to both sides

- 0.5 + cos (x) + 0.5 = 0 + 0.5

Simplify

cos (x) = 0.5

cos (x) = 0.5

General solutions for

cos (x) = 0.5

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

sin (2 x) tan (2 x + 2) = sin (3 x - 1) tan (2 x + 2)

2 sin^{2} (θ) - 7 sin (θ) - 4 = 0

2 tan (x) - 23 = 0

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

2 sin^{2} (\frac{x}{4}) - 3 cos (\frac{x}{4}) = 0

What is the general solution for sin(x)cot(x)=0.5 ?
The general solution for sin(x)cot(x)=0.5 is x= pi/3+2pin,x=(5pi)/3+2pin