1. Compound Angles

hi people i have a question on compound angles which im finding quite annoying could someone show me how to solve the following questions:

express the following

1. 6sinwt - 8coswt in the form of: R sin(Θ - a)

2. 10sinΘ + 12sin(Θ+30°) in the form of: R sin (Θ + a°)

2. Re: Compound Angles

You know that $\displaystyle 3^2+4^2 = 5^2$, so $\displaystyle 6^2+8^2 = 10^2$. Hence, for #1, factor out 10:

$\displaystyle 10\left(\dfrac{3}{5} \sin (wt) - \dfrac{4}{5} \cos(wt) \right)$

There exists an angle $\displaystyle a$ such that $\displaystyle \cos a = \dfrac{3}{5}$ and $\displaystyle \sin a = \dfrac{4}{5}$ (it is the angle adjacent to the sides of length 3 and 5 in a 3-4-5 right triangle).

So, you have $\displaystyle 10(\sin(wt)\cos(a) - \cos(wt)\sin(a)) = 10\sin(wt - a)$.

Do something similar for #2.

3. Re: Compound Angles

Thanks slipeternal for replying i understand question 1 now but im still confused on the second part how would i implement that method in the second part.

4. Re: Compound Angles

Use the sum of angles formula to expand $\displaystyle \sin(\theta+30^\circ)$:

\displaystyle \begin{align*}10\sin \theta + 12\sin(\theta+30^\circ) & = 10 \sin \theta + 12(\sin \theta \cos 30^\circ + \cos \theta \sin 30^\circ) \\ & = (10+12\cos 30^\circ)\sin \theta + (12\sin 30^\circ)\cos \theta\end{align*}

Evaluate $\displaystyle \sin 30^\circ$ and $\displaystyle \cos 30^\circ$ and then find a triangle whose sides have lengths $\displaystyle 10+12\cos 30^\circ$ and $\displaystyle 12\sin 30^\circ$.