What is the general solution for 1sin(45)=2.42sin(r) ?

The general solution for 1sin(45)=2.42sin(r) is r=0.29651…+360n,r=180-0.29651…+360n

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1sin(45)=2.42sin(r)

1 sin (4 5^{\circ}) = 2.42 sin (r)

r = 0.29651 \dots + 36 0^{\circ} n, r = 18 0^{\circ} - 0.29651 \dots + 36 0^{\circ} n

Radians

r = 0.29651 \dots + 2 πn, r = π - 0.29651 \dots + 2 πn

Solution steps

1 \cdot sin (4 5^{\circ}) = 2.42 sin (r)

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 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}

1 \cdot \frac{2}{2} = 2.42 sin (r)

Switch sides

2.42 sin (r) = 1 \cdot \frac{2}{2}

Multiply both sides by

100

2.42 sin (r) = 1 \cdot \frac{2}{2}

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

2.42 sin (r) \cdot 100 = 1 \cdot \frac{2}{2} \cdot 100

Refine

242 sin (r) = 502

242 sin (r) = 502

Divide both sides by

242

242 sin (r) = 502

Divide both sides by

242

\frac{242 sin ( r )}{242} = \frac{50 2}{242}

Simplify

sin (r) = \frac{25 2}{121}

sin (r) = \frac{25 2}{121}

Apply trig inverse properties

sin (r) = \frac{25 2}{121}

General solutions for

sin (r) = \frac{25 2}{121}

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

r = arcsin (\frac{25 2}{121}) + 36 0^{\circ} n, r = 18 0^{\circ} - arcsin (\frac{25 2}{121}) + 36 0^{\circ} n

r = arcsin (\frac{25 2}{121}) + 36 0^{\circ} n, r = 18 0^{\circ} - arcsin (\frac{25 2}{121}) + 36 0^{\circ} n

Show solutions in decimal form

r = 0.29651 \dots + 36 0^{\circ} n, r = 18 0^{\circ} - 0.29651 \dots + 36 0^{\circ} n

What is the general solution for 1sin(45)=2.42sin(r) ?
The general solution for 1sin(45)=2.42sin(r) is r=0.29651…+360n,r=180-0.29651…+360n