## Task

In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when combined with relations you have already learned.

For the following task, assume that the sum angle formula for sine is true.
Namely,
$$
\sin(\theta+\phi)= \sin\theta \cos\phi +\cos\theta \sin\phi.
$$

- To derive the difference angle formula for sine, write $\sin(\theta-\phi)$ as $\sin(\theta+(-\phi))$ and apply the sum angle formula for sine to the angles $\theta$ and $-\phi$. Use the fact that sine is an odd function while cosine is even function to simplify your answer.
Conclude that
$$
\sin(\theta-\phi) = \sin(\theta) \cos(\phi)-\cos(\theta)\sin(\phi).
$$
- To derive the sum angle formula for cosine, use what what you learned in (a) to show that
$$
\cos(\theta+\phi)= \cos\theta \cos\phi -\sin\theta \sin\phi.
$$
You may want to start with an exploration of $\sin\left(\frac{\pi}{2} -(\theta+\phi)\right)$.
- Derive the difference angle formula for cosine,
$$
\cos(\theta-\phi)= \cos\theta \cos\phi +\sin\theta \sin\phi.
$$
- Derive the sum angle formula for tangent,
$$
\tan(\theta+\phi) = \frac{\tan\theta+\tan\phi}{1-\tan\theta\tan\phi}.
$$
- Derive the difference angle formula for tangent,
$$
\tan(\theta-\phi) = \frac{\tan\theta-\tan\phi}{1+\tan\theta\tan\phi}.
$$

## IM Commentary

The goal of this task is to have students derive the addition and subtraction formulas for cosine and tangent, and the subtraction formula for cosine, from the sum formula for sine. The task provides varying levels of scaffolding, pointing out possible relations to use early on, but leaving more creative work for the student later. In addition, the task assumes the sum angle formula for sine and shows how the other sum and difference formulas must follow.

This text of this problem and its solution assumes familiarity with the Greek letters theta $(\theta)$ and phi $(\phi)$. However, some teachers or books will use alpha $(\alpha)$ and beta $(\beta)$. Still others use Latin letters like $u$ and $v$ or $A$ and $B$. Instructors should feel free to change the letters to match those of their source, as the choice of letters is not important; it is the relationships the letters represent that are useful.

Before embarking on this task, students should be aware that sine is odd (hence $\sin(-\theta)=-\sin(\theta))$ and cosine is even (hence $\cos(-\theta)=\cos(\theta))$ as found in standard F-TF.4. Students should know the relations between sine, cosine and tangent found in standard G-SRT.6. In addition, students should know the relation between trigonometric values of "complementary" angles found in standard G-SRT.7, ($\sin(\theta) = \cos(\pi/2-\theta)$, etc.).

The emphasis of this task is to show how one result can be extended to a family of results using known relations. This is a central strategy in mathematical thinking, and illustrates Standards for Mathematical Practices 7 and 8, looking for structure and making use of repeated reasonin. Along these lines, solutions other than the ones given here are also viable -- for example, students might prove part (e) from part (d) by making the "substitution" $\phi\to -\phi$ (perhaps not in that language).