Key to "Solving sin⁡(arcsin⁡(a−x)−c)=sin⁡ for x (arcsin⁡(b+x)+π−c)+d"

maliaseth0

Answered question

2022-01-24

Solving

\sin (\arcsin (a - x) - c) = \sin

for x

(\arcsin (b + x) + π - c) + d

Hana Larsen

Beginner2022-01-25Added 17 answers

Use the identity

\sin (x + y) = \sin x \cos y + \cos x \sin y

to expand the equation,
to

(a - x) \cos c - \sqrt{1 - {(a - x)}^{2}} \sin c = - (b + x) \cos c + \sqrt{1 - {(b + x)}^{2}} \sin c + d

where

\cos (\arcsin t) = \sqrt{1 - t^{2}}

is used. Rearrange the resulting equation to obtain the following,

\sqrt{1 - {(a - x)}^{2}} = r - \sqrt{1 - {(b + x)}^{2}}

where

r = \frac{(a + b) \cos c - d}{\sin c}

Then, square both sides (twice) to remove square-roots and you end up with a standard quadratic equation in x.

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