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Popular Trigonometry >

solvefor x,sin(xθ)= 1/2

Solution

solvefor

Solution

x = \frac{π}{6 θ} + \frac{2 πn}{θ}, x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

Solution steps

sin (x θ) = \frac{1}{2}

General solutions for

sin (x θ) = \frac{1}{2}

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 θ = \frac{π}{6} + 2 πn, x θ = \frac{5 π}{6} + 2 πn

x θ = \frac{π}{6} + 2 πn, x θ = \frac{5 π}{6} + 2 πn

Solve

x θ = \frac{π}{6} + 2 πn : x = \frac{π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x θ = \frac{π}{6} + 2 πn

Divide both sides by

θ; θ \neq = 0

x θ = \frac{π}{6} + 2 πn

Divide both sides by

θ; θ \neq = 0

\frac{x θ}{θ} = \frac{\frac{π}{6}}{θ} + \frac{2 πn}{θ}; θ \neq = 0

Simplify

\frac{x θ}{θ} = \frac{\frac{π}{6}}{θ} + \frac{2 πn}{θ}

Simplify

\frac{x θ}{θ} : x

\frac{x θ}{θ}

Cancel the common factor:

θ

= x

Simplify

\frac{\frac{π}{6}}{θ} + \frac{2 πn}{θ} : \frac{π}{6 θ} + \frac{2 πn}{θ}

\frac{\frac{π}{6}}{θ} + \frac{2 πn}{θ}

Apply the fraction rule:

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

= \frac{π}{6 θ} + \frac{2 πn}{θ}

x = \frac{π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x = \frac{π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x = \frac{π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

Solve

x θ = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x θ = \frac{5 π}{6} + 2 πn

Divide both sides by

θ; θ \neq = 0

x θ = \frac{5 π}{6} + 2 πn

Divide both sides by

θ; θ \neq = 0

\frac{x θ}{θ} = \frac{\frac{5 π}{6}}{θ} + \frac{2 πn}{θ}; θ \neq = 0

Simplify

\frac{x θ}{θ} = \frac{\frac{5 π}{6}}{θ} + \frac{2 πn}{θ}

Simplify

\frac{x θ}{θ} : x

\frac{x θ}{θ}

Cancel the common factor:

θ

= x

Simplify

\frac{\frac{5 π}{6}}{θ} + \frac{2 πn}{θ} : \frac{5 π}{6 θ} + \frac{2 πn}{θ}

\frac{\frac{5 π}{6}}{θ} + \frac{2 πn}{θ}

Apply the fraction rule:

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

= \frac{5 π}{6 θ} + \frac{2 πn}{θ}

x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

x = \frac{π}{6 θ} + \frac{2 πn}{θ}, x = \frac{5 π}{6 θ} + \frac{2 πn}{θ}; θ \neq = 0

Graph

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