What is the general solution for sin(x)=(11sin(120))/(18) ?

The general solution for sin(x)=(11sin(120))/(18) is x=0.55770…+360n,x=180-0.55770…+360n

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sin(x)=(11sin(120))/(18)

sin (x) = \frac{11 sin ( 12 0 ^{\circ} )}{18}

x = 0.55770 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.55770 \dots + 36 0^{\circ} n

Radians

x = 0.55770 \dots + 2 πn, x = π - 0.55770 \dots + 2 πn

Solution steps

sin (x) = \frac{11 sin ( 12 0 ^{\circ} )}{18}

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\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{3}{2}

= \frac{3}{2}

sin (x) = \frac{11 \cdot \frac{3}{2}}{18}

Simplify

\frac{11 \cdot \frac{3}{2}}{18} : \frac{11 3}{36}

\frac{11 \cdot \frac{3}{2}}{18}

Multiply

11 \cdot \frac{3}{2} : \frac{11 3}{2}

11 \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 \cdot 11}{2}

= \frac{\frac{11 3}{2}}{18}

Apply the fraction rule:

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

= \frac{3 \cdot 11}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{11 3}{36}

Apply trig inverse properties

sin (x) = \frac{11 3}{36}

General solutions for

sin (x) = \frac{11 3}{36}

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

x = arcsin (\frac{11 3}{36}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{11 3}{36}) + 36 0^{\circ} n

x = arcsin (\frac{11 3}{36}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{11 3}{36}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.55770 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.55770 \dots + 36 0^{\circ} n

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

8 cos^{2} (x) - 10 sin (x) - 11 = 0

What is the general solution for sin(x)=(11sin(120))/(18) ?
The general solution for sin(x)=(11sin(120))/(18) is x=0.55770…+360n,x=180-0.55770…+360n