How to Solve asinx + bcosx = c: Understanding the Procedure and Limitations

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To solve the equation asin(x) + bcos(x) = c, first determine R as the square root of a^2 + b^2. This allows the equation to be rewritten as Rsin(x + w), where tan(w) = b/a. The values of a and b can be positive or negative, but the relationship a^2 + b^2 = 1 is not always valid, which is why R is defined. When calculating w using the inverse tangent, any value can be used, but the smallest positive value is typically preferred for simplicity. Understanding these steps clarifies the procedure and addresses limitations in solving the equation.
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hello!

I would like to know how to solve asinx + bcosx = c, with a, b, c being any real numbers (constants)

First, are there any limitations for the above to be valid?

Second, I was introduced to a solution but I cannot fully understand the procedure.

Let's say we have asinx + bcosx = c
We can solve this by using:
R= root of a^2 plus b^2
and
asinx + bcosx = Rsin(x+w)
and
tan(w)=b over a

the first question is, do the above are valid for a, b being either positive or negative?

Second, when we find the "w" using the calculator (by inversing its tan), how do we find which exactly value of w we must use?

Third, when we inverse the sin(x+w), by the calculator, how do we find which exactly values of x+w we must use?

Thanks!
 
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That comes from the "sum formula" that says that sin(x+ w)= sin(x)cos(w)+ cos(x)sin(w). If we are given asin(x)+ bcos(x), comparing to the previous formula, we would like to have a= cos(w) and b= sin(w). But that is not always possible, because that would require that a^2+ b^2= cos^2(w)+ sin^2(w)= 1 which is of course, not always true! But if we define R= \sqrt{a^2+ b^2} we can "multiply and divide" by R: asin(x)+ bcos(x)= R((a/R)sin(x)+ (b/R)cos(x)) and now (a/R)^2+ (b/R)^2= \frac{a^2+ b^2}{R^2}= \frac{a^2+ b^2}{a^2+ b^2}= 1. So we can say cos(w)= a/R and sin(w)= b/R so that w= arccos(a/R)= arcsin(b/R). Or, since cos(w)= a/R and sin(w)= b/R, tan(w)= sin(w)/cos(w)= (b/R)/(a/R)= b/a..

You ask: "when we find the "w" using the calculator (by inversing its tan), how do we find which exactly value of w we must use?"
Any if the values can be used. Use whichever fits your needs. Typically, that is the smallest positive value just because it is simplest.
 
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