What topic does this equation $ 3\sin x -4\cos x = a$ belong to?

Question

Question : Find the values of the parameter a for which the equation $ 3\sin x -4\cos x = a$ has a solution.

Not looking for the answer here but can somebody please recommend me a book or a website to get sufficient knowledge on this topic so that I'm able to answer this question? (I'm familiar with parametric equations involving only unknowns, but this is my first time seeing trigonometric functions in one)

Thanks in advance.

https://www.quora.com/What-is-the-maximum-value-of-y-Asinx+Bcosx-where-and-b-are-positive-constants-A-B,does this help? — Albus Dumbledore, Sep 05 '20 at 13:56
@SarGe : Correct me if I'm wrong but after some calculation, I got $ 5sin(x+b)=a$. So because both $sinx$ and $cosx$ has to be between -1 and 1, 5 is multiplied with both of these numbers to get all values of the parameter a. Is this correct? — ThermalRaindrops62, Sep 05 '20 at 14:52
Use the derivative of $f(x)=3\sin x -4\cos x -a$ to find the minimum and maximum values of $f(x)$ — Claude Leibovici, Sep 05 '20 at 16:19
The expression $3\sin x - 4\cos x$ is a linear combination of sines and cosines. — N. F. Taussig, Sep 06 '20 at 09:49

score 0 · Answer 1 · answered Sep 05 '20 at 14:28

0

$$3 \sin x-4 \cos x=a \implies (3/5) \sin x-(4/5) \cos x=a/5\implies \sin (x-t)=a/5, \tan t=4/3~~~~(1)$$ This equation will be valid if $-5 \le a\le 5.$ Solutions of (1) are given by $$x= n\pi+(-1)^n \sin^{-1}(a/5)+t, n \in I$$

answered Sep 05 '20 at 14:28

Z Ahmed

43,235

Can explain how you get $-5 \le a\le 5$ ? – ThermalRaindrops62 Sep 05 '20 at 14:38
@ThermalRaindrops62, see the link that I provided. – SarGe Sep 05 '20 at 14:40

What topic does this equation $ 3\sin x -4\cos x = a$ belong to?

1 Answers1