3.5 - Angle Sum & Difference Formulas

Prior Lesson

Objectives:

• Understand and use the half-angle, angle sum, and angle difference formulas.

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Lesson Guide

Half-Angle Formulas

Recall that as part of Problem 3 on worksheet 12.1, students developed two alternate forms of the double-angle formula for cosine:

• $\cos(2\theta)=2\cos^2(\theta)-1$, and
• $\cos(2\theta)=1-2\sin^2(\theta)$.

By relabeling these quantities (i.e., let $\phi=2\theta$ be our "original" angle, and $\phi/2=\theta$ the "original" angle divided by $2$), we equivalently have:

• $\cos(\phi)=2\cos^2(\phi/2)-1$, and
• $\cos(\phi)=1-2\sin^2(\phi/2)$.

Rewriting these equations, we obtain:

 $\cos(\phi/2)=\pm\sqrt{\frac{1+\cos(\phi)}{2}}$ and $\sin(\phi/2)=\pm\sqrt{\frac{1-\cos(\phi)}{2}}$.
 These are called the \underline{half-angle formulas} for cosine and sine, respectively.

In both half-angle formulas, we have a choice: should we use the positive or negative square root? The answer will depend on the quadrant the angle lies in.

Motivate the usefulness of these formulas by telling students that we know the exact values of $\sin\theta$ and $\cos\theta$ for relatively few values of $\theta$, but with the half-angle formulas, we can find more. Give an example or two, demonstrating how we might use the half-angle formulas. For example, you could ask students to find $\sin(\pi/8)$. Make sure to emphasize how to decide whether to choose the positive or negative square root.

Have students complete Problem 1 on the worksheet.

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Angle Sum and Difference Formulas

 $\sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \sin(\phi)\cos(\theta)$
 $\sin(\theta - \phi) = \sin(\theta)\cos(\phi) - \sin(\phi)\cos(\theta)$
 $\cos(\theta + \phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)$
 $\cos(\theta - \phi) = \cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi)$

These formulas, again, help us to find the exact sine and cosine of an even larger assortment of angles. Give an example of how one might use these formulas (for example, can students find $\cos(345^{\circ})$?).

Have students complete the worksheet.

Comments

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