What is the general solution for 2/3 =(sin(x))/(sin(135)) ?

The general solution for 2/3 =(sin(x))/(sin(135)) is x=0.49088…+360n,x=180-0.49088…+360n

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2/3 =(sin(x))/(sin(135))

\frac{2}{3} = \frac{sin ( x )}{sin ( 13 5 ^{\circ} )}

x = 0.49088 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.49088 \dots + 36 0^{\circ} n

Radians

x = 0.49088 \dots + 2 πn, x = π - 0.49088 \dots + 2 πn

Solution steps

\frac{2}{3} = \frac{sin ( x )}{sin ( 13 5 ^{\circ} )}

Switch sides

\frac{sin ( x )}{sin ( 13 5 ^{\circ} )} = \frac{2}{3}

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2}

\frac{sin ( x )}{\frac{2}{2}} = \frac{2}{3}

Multiply both sides by

\frac{2}{2}

\frac{sin ( x )}{\frac{2}{2}} = \frac{2}{3}

Multiply both sides by

\frac{2}{2}

\frac{sin ( x ) \frac{2}{2}}{\frac{2}{2}} = \frac{2 \cdot \frac{2}{2}}{3}

Simplify

\frac{sin ( x ) \frac{2}{2}}{\frac{2}{2}} = \frac{2 \cdot \frac{2}{2}}{3}

Simplify

\frac{sin ( x ) \frac{2}{2}}{\frac{2}{2}} : sin (x)

\frac{sin ( x ) \frac{2}{2}}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{sin ( x ) \frac{2}{2} \cdot 2}{2}

Multiply

sin (x) \frac{2}{2} \cdot 2 : 2 sin (x)

sin (x) \frac{2}{2} \cdot 2

Multiply fractions:

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

= \frac{2 sin ( x ) \cdot 2}{2}

Cancel the common factor:

2

= 2 sin (x)

= \frac{2 sin ( x )}{2}

Cancel the common factor:

2

= sin (x)

Simplify

\frac{2 \cdot \frac{2}{2}}{3} : \frac{2}{3}

\frac{2 \cdot \frac{2}{2}}{3}

Multiply

2 \cdot \frac{2}{2} : 2

2 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 2}{2}

Cancel the common factor:

2

= 2

= \frac{2}{3}

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{2}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2}{3}) + 36 0^{\circ} n

x = arcsin (\frac{2}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2}{3}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.49088 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.49088 \dots + 36 0^{\circ} n

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

a sin (2 θ) = 0

cos (2 x) = 0, x \leq 2 π, 0

7 + csc (2 x) = 3 cot^{2} (2 x)

cos (θ) = cos (\frac{π}{13})

What is the general solution for 2/3 =(sin(x))/(sin(135)) ?
The general solution for 2/3 =(sin(x))/(sin(135)) is x=0.49088…+360n,x=180-0.49088…+360n