My book states that:
The formula for the cosine of the difference of two angles is deduced as an application of the scalar product of two vectors:
$$\cos(\alpha - \beta) = \cos\alpha\cos\beta+\sin\alpha\sin\beta$$
From this formula, we can deduce the formula for the sine of the difference:
$$\sin(\alpha-\beta)=\cos[\frac{\pi}{2}-(\alpha-\beta)]=\\\cos[\frac{\pi}{2}-\alpha-(-\beta)] =\\ \cos(\frac{\pi}{2}-\alpha)\cos(-\beta)+\sin(\frac{\pi}{2}-\alpha)\sin(-\beta)$$
$$\\\\$$ $$\\\\\\\\\\\\\sin(\alpha-\beta )= \sin\alpha\cos\beta-\cos\alpha\sin\beta$$
Deduce the following expression starting from the formulas above:
- $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$
I have two questions:
- I don't understand how my book means by
The formula for the cosine of the difference of two angles is deduced as an application of the scalar product of two vectors:
Could you explain to me what this means?
- How do I solve the given problem?