# Using constraints when solving a trigometric equation [closed]

I'm trying to learn how to add constraints/assumptions to equations when using Solve, sp as to avoid multi-valued solutions. For example, I'm trying to use Mathematica 11.3 to solve the equation

n Sin[theta] == Sin[alpha]


for theta, where I know n > 1 and that the solution is in the first quadrant; i.e., 0 < theta < Pi/2).

I tried

Solve[{n Sin[theta] == Sin[alpha], alpha > 0, alpha < π/4, θ > 0, θ < π/2}, theta]


and got the solution:

{{theta ->
Conditional
Expression[π - ArcSin[Sin[alpha]/n] + 2 π C[1],
C[1] ∈ Integers && 0 < alpha < π/4 && 0 < θ < π/2]},
{theta ->
ConditionalExpression[
ArcSin[Sin[alpha]/n] + 2 π C[1],
C[1] ∈ Integers && 0 < alpha < π/4 && 0 < θ < π/2]}}


which seems needlessly complex to me. I tried to simplify this solution, as suggested here: but it didn't simplify. I was hoping it would reduce to

ArcSin[Sin[alpha]/n]


## closed as off-topic by m_goldberg, MarcoB, Bob Hanlon, bbgodfrey, José Antonio Díaz NavasMar 21 at 19:50

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – m_goldberg, MarcoB, Bob Hanlon, bbgodfrey, José Antonio Díaz Navas
If this question can be reworded to fit the rules in the help center, please edit the question.

I think the problem is that θ is not the same as theta. Fixing that, and also adding your constraint on n:
Solve[

{{theta -> ConditionalExpression[ArcSin[Sin[alpha]/n], 0 < alpha < π/4 && n > 1]}}

• Assuming[assume =0 < alpha < Pi/4 && 0 < theta < Pi/2 && n > 1, Solve[n Sin[theta] == Sin[alpha] && assume, theta] // Simplify][[1]] – Bob Hanlon Mar 19 at 4:40