I have been trying to understand this problem from The Theoretical Minimum. The following solution is given for the equation of motion of a harmonic oscillator.

$$

\vec{r}(t)=\vec{c}_1\cos(\omega t)+\vec{c}_2\sin(\omega t)

$$

Its rewritten for a given angle $\theta$.

$$

\vec{r}(t)=\vec{b}_1\cos(\omega t – \theta)+\vec{b}_2\sin(\omega t – \theta)

$$

Where

$$

\vec{b}_1=\vec{c}_1\cos(\theta)+\vec{c}_2\sin(\theta)

$$

and

$$

\vec{b}_2=\vec{c}_2\cos(\theta)-\vec{c}_1\sin(\theta)

$$

I want to know what identity or trigonometric property is this.

Thanks.

## Best Answer

They are exploiting the following identities:

Use the last two on the equation: $$\vec{r}(t)=\vec{b}_1\cos(\omega t - \theta) + \vec{b}_2\sin(\omega t - \theta)$$ expand it and then group identical terms of sine and cosine of $\omega t$, and you should reach the first version independent of $\theta$ (well, is just a constant so it still hiden into the other constants).