What is the general solution for 1.65*sin(30)=1.33*sin(x) ?

The general solution for 1.65*sin(30)=1.33*sin(x) is x=0.66912…+360n,x=180-0.66912…+360n

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1.65sin(30)=1.33sin(x)

1.65 \cdot sin (3 0^{\circ}) = 1.33 \cdot sin (x)

x = 0.66912 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.66912 \dots + 36 0^{\circ} n

Radians

x = 0.66912 \dots + 2 πn, x = π - 0.66912 \dots + 2 πn

Solution steps

1.65 sin (3 0^{\circ}) = 1.33 sin (x)

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

= \frac{1}{2}

1.65 \cdot \frac{1}{2} = 1.33 sin (x)

Switch sides

1.33 sin (x) = 1.65 \cdot \frac{1}{2}

Divide both sides by

1.33

1.33 sin (x) = 1.65 \cdot \frac{1}{2}

Divide both sides by

1.33

\frac{1.33 sin ( x )}{1.33} = \frac{1.65 \cdot \frac{1}{2}}{1.33}

Simplify

\frac{1.33 sin ( x )}{1.33} = \frac{1.65 \cdot \frac{1}{2}}{1.33}

Simplify

\frac{1.33 sin ( x )}{1.33} : sin (x)

\frac{1.33 sin ( x )}{1.33}

Cancel the common factor:

1.33

= sin (x)

Simplify

\frac{1.65 \cdot \frac{1}{2}}{1.33} : 0.62030 \dots

\frac{1.65 \cdot \frac{1}{2}}{1.33}

Multiply

1.65 \cdot \frac{1}{2} : 0.825

1.65 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 1.65}{2}

Multiply the numbers:

1 \cdot 1.65 = 1.65

= \frac{1.65}{2}

Divide the numbers:

\frac{1.65}{2} = 0.825

= 0.825

= \frac{0.825}{1.33}

Divide the numbers:

\frac{0.825}{1.33} = 0.62030 \dots

= 0.62030 \dots

sin (x) = 0.62030 \dots

sin (x) = 0.62030 \dots

sin (x) = 0.62030 \dots

Apply trig inverse properties

sin (x) = 0.62030 \dots

General solutions for

sin (x) = 0.62030 \dots

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

x = arcsin (0.62030 \dots) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (0.62030 \dots) + 36 0^{\circ} n

x = arcsin (0.62030 \dots) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (0.62030 \dots) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.66912 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.66912 \dots + 36 0^{\circ} n

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

cos (2 x) - sin (x) = 0.5

cos (θ) - 2 sin^{2} (θ) = - 1

4 + 4 cos (θ) = 0

- cot (2 θ) = 0

What is the general solution for 1.65*sin(30)=1.33*sin(x) ?
The general solution for 1.65*sin(30)=1.33*sin(x) is x=0.66912…+360n,x=180-0.66912…+360n