What is the general solution for 60sin(45)=400sin(θ) ?

The general solution for 60sin(45)=400sin(θ) is θ=0.10626…+360n,θ=180-0.10626…+360n

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60sin(45)=400sin(θ)

60 sin (4 5^{\circ}) = 400 sin (θ)

θ = 0.10626 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.10626 \dots + 36 0^{\circ} n

Radians

θ = 0.10626 \dots + 2 πn, θ = π - 0.10626 \dots + 2 πn

Solution steps

60 sin (4 5^{\circ}) = 400 sin (θ)

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}

60 \cdot \frac{2}{2} = 400 sin (θ)

Switch sides

400 sin (θ) = 60 \cdot \frac{2}{2}

Multiply fractions:

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

400 sin (θ) = \frac{2 \cdot 60}{2}

Divide the numbers:

\frac{60}{2} = 30

400 sin (θ) = 302

Divide both sides by

400

400 sin (θ) = 302

Divide both sides by

400

\frac{400 sin ( θ )}{400} = \frac{30 2}{400}

Simplify

sin (θ) = \frac{3 2}{40}

sin (θ) = \frac{3 2}{40}

Apply trig inverse properties

sin (θ) = \frac{3 2}{40}

General solutions for

sin (θ) = \frac{3 2}{40}

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

θ = arcsin (\frac{3 2}{40}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{3 2}{40}) + 36 0^{\circ} n

θ = arcsin (\frac{3 2}{40}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{3 2}{40}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 0.10626 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.10626 \dots + 36 0^{\circ} n

(1 + tan (x)) (5 sin (x) - 2) = 0

arccos (e - x) = π^{3}

sin (a) = \frac{20}{25}

1 + cos (x) = 3 - sin (x)

3 = 7 cos (\frac{π}{3} t)

What is the general solution for 60sin(45)=400sin(θ) ?
The general solution for 60sin(45)=400sin(θ) is θ=0.10626…+360n,θ=180-0.10626…+360n