Pre-Calculus Test Questions: Sum and Difference Identities for Sine, Cosine, Tangent

📚 Quick Study Guide

• 📐 Sine Sum and Difference:
  • ➕ $\sin(a + b) = \sin a \cos b + \cos a \sin b$
  • ➖ $\sin(a - b) = \sin a \cos b - \cos a \sin b$
• 🧮 Cosine Sum and Difference:
  • ➕ $\cos(a + b) = \cos a \cos b - \sin a \sin b$
  • ➖ $\cos(a - b) = \cos a \cos b + \sin a \sin b$
• ➗ Tangent Sum and Difference:
  • ➕ $\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$
  • ➖ $\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}$
• 💡 Key Tip: Remember the sign changes! Cosine's sum formula has a subtraction, and difference has addition. Tangent formulas involve fractions.
• ✍️ Common Values: Know your unit circle! Values like $\sin(\frac{\pi}{6})$, $\cos(\frac{\pi}{4})$, and $\tan(\frac{\pi}{3})$ are crucial.

✍️ Practice Quiz

1. What is the value of $\sin(75^\circ)$?
   1. $\frac{\sqrt{6} + \sqrt{2}}{4}$
   2. $\frac{\sqrt{6} - \sqrt{2}}{4}$
   3. $\frac{\sqrt{2} + \sqrt{2}}{2}$
   4. $\frac{\sqrt{3} - 1}{2}$
2. Simplify $\cos(x + y) + \cos(x - y)$.
   1. $2\cos x \cos y$
   2. $2\sin x \sin y$
   3. $0$
   4. $2\sin x \cos y$
3. What is the value of $\tan(15^\circ)$?
   1. $2 - \sqrt{3}$
   2. $2 + \sqrt{3}$
   3. $\sqrt{3} - 2$
   4. $1$
4. Find the value of $\sin(a - b)$ if $\sin a = \frac{3}{5}$, $\cos a = \frac{4}{5}$, $\sin b = \frac{5}{13}$, and $\cos b = \frac{12}{13}$.
   1. $\frac{16}{65}$
   2. $\frac{63}{65}$
   3. $\frac{33}{65}$
   4. $\frac{56}{65}$
5. Simplify $\frac{\tan 70^\circ - \tan 40^\circ}{1 + \tan 70^\circ \tan 40^\circ}$.
   1. $\sqrt{3}$
   2. $\frac{\sqrt{3}}{3}$
   3. $1$
   4. $0$
6. If $\cos(x + y) = \frac{1}{2}$ and $\sin x = \frac{1}{3}$, $\sin y = \frac{1}{4}$, find $\cos x \cos y$.
   1. $\frac{7}{12}$
   2. $\frac{5}{12}$
   3. $\frac{1}{12}$
   4. $\frac{11}{12}$
7. What is the value of $\cos(\frac{\pi}{12})$?
   1. $\frac{\sqrt{6} + \sqrt{2}}{4}$
   2. $\frac{\sqrt{6} - \sqrt{2}}{4}$
   3. $\frac{\sqrt{2} + \sqrt{2}}{2}$
   4. $\frac{\sqrt{3} - 1}{2}$