trigonometric way to solve Sine and Cosine sums?

Question

drawing graphs for most addition or subtractions of Sine and Cosine formula lead to another sine or cosine shaped graph, but I don't know how to actually write them as a formula.
For example , drawing plots of " $ a\ cos(\theta) -b \ sin(\theta) $ " and " $ a \ sin(\theta) + b \ cos(\theta) $ " both yield as sine-shaped graphs which means one should be able to write them in a form like :

$ a\ cos(\theta) -b \ sin(\theta) = A \ sin(\phi)$

or

$ a \ sin(\theta) + b \ cos(\theta) = B \ cos(\omega) $

but I can't think of a rational way of finding $ A,B, \phi \ and \ \omega$. I highly appreciate if someone can shed light on this for me.

Answer 1

If they are written with the same trigonometric function, you could find the value you are looking for by adding complex numbers.

First note that $$\cos \theta = \sin \left(\theta + \frac\pi2\right)$$

If you want to find the sum of two sine funtions $$a_1\sin(\theta + \phi_1) + a_2\sin(\theta + \phi_2)$$ you sum the complex numbers $a_1e^{i\phi_1}$ and $a_2e^{i\phi_2}$ $$a_1e^{i\phi_1} + a_2e^{i\phi_2} = Ae^{i\omega}$$ Then $$a_1\sin(\theta + \phi_1) + a_2\sin(\theta + \phi_2) = A\sin(\theta +\omega)$$

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Here an example $$4\sin\theta + 3\cos\theta$$ We express both of them as sine $$4\sin\theta + 3\sin\left(\theta+\frac\pi2\right)$$ Then, we sum the complex numbers $4e^{i0}$ and $3e^{i\frac\pi2}$ \begin{align}4 + 3e^{i\frac\pi2} &= 4 + 3\cos \frac\pi2 + 3i\sin\frac\pi2\\ &= 4 + 3i\\ &= 5e^{i\arctan\left(\frac34\right)}\end{align} Finally $$4\sin\theta+3\cos\theta = 5\sin\left(\theta+\arctan\left(\frac34\right)\right)$$

Answer 2

Hint Expand using the angle sum identity for $\sin$ i.e., write $$a \sin \theta + b \cos \theta = A \sin(\theta_0 + \theta) = A \cos \theta_0 \sin \theta + A \sin \theta_0 \cos \theta .$$ Now, match coefficients of $\sin \theta$, $\cos \theta$ to get $$a = A \cos \theta_0, \qquad b = A \sin \theta_0 .$$ Solving for $A, \theta_0$ is now essentially converting from Cartesian to polar coordinates.

Answer 3

Hint

$$ \sin ( \alpha + \beta ) = \sin \alpha \cdot \cos \beta + \cos \alpha\cdot \sin \beta $$

Make a triangle with given constants a and b and use one of its angles to arrive at the answer. You could also use the formula for $ \cos (\alpha + \beta) $ to arrive at a result in terms of cos.