๐ Understanding Sum and Difference Identities
Sum and difference identities are trigonometric identities that allow you to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. They are essential tools in pre-calculus for simplifying expressions, solving equations, and proving other identities. Let's dive in!
โโ Definition of Sum Identities
Sum identities express trigonometric functions of the sum of two angles, say $A$ and $B$, in terms of trigonometric functions of $A$ and $B$ individually.
๐๐งญ Definition of Difference Identities
Difference identities express trigonometric functions of the difference of two angles, say $A$ and $B$, in terms of trigonometric functions of $A$ and $B$ individually.
๐ Sum vs. Difference Identities: A Comparison
| Feature |
Sum Identities |
Difference Identities |
| Sine |
$\sin(A + B) = \sin A \cos B + \cos A \sin B$ |
$\sin(A - B) = \sin A \cos B - \cos A \sin B$ |
| Cosine |
$\cos(A + B) = \cos A \cos B - \sin A \sin B$ |
$\cos(A - B) = \cos A \cos B + \sin A \sin B$ |
| Tangent |
$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ |
$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ |
| Operation |
Deals with the addition of angles. |
Deals with the subtraction of angles. |
๐ Key Takeaways
- ๐งฎ Remember the formulas: Accurate application is crucial for solving problems.
- ๐ Practice substitution: The more you practice, the easier it will be to recognize when and how to apply these identities.
- ๐ง Be mindful of signs: A small sign error can lead to a completely wrong answer.
๐ Example Problems Solved
Let's walk through some examples to see these identities in action.
โ๏ธ Problem 1: Simplify $\sin(75^{\circ})$
We can write $75^{\circ}$ as $45^{\circ} + 30^{\circ}$. Using the sine sum identity:
$\sin(75^{\circ}) = \sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}$
$= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = \frac{\sqrt{6} + \sqrt{2}}{4}$
๐ Problem 2: Simplify $\cos(15^{\circ})$
We can write $15^{\circ}$ as $45^{\circ} - 30^{\circ}$. Using the cosine difference identity:
$\cos(15^{\circ}) = \cos(45^{\circ} - 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} + \sin 45^{\circ} \sin 30^{\circ}$
$= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = \frac{\sqrt{6} + \sqrt{2}}{4}$
๐งฎ Problem 3: Simplify $\tan(105^{\circ})$
We can write $105^{\circ}$ as $60^{\circ} + 45^{\circ}$. Using the tangent sum identity:
$\tan(105^{\circ}) = \tan(60^{\circ} + 45^{\circ}) = \frac{\tan 60^{\circ} + \tan 45^{\circ}}{1 - \tan 60^{\circ} \tan 45^{\circ}}$
$= \frac{\sqrt{3} + 1}{1 - \sqrt{3}} = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}$
๐ค Practice Quiz
Test your knowledge with these practice questions!
- โ Simplify $\sin(15^{\circ})$
- โ Simplify $\cos(75^{\circ})$
- โ Simplify $\tan(15^{\circ})$
- โ Find the exact value of $\sin(165^{\circ})$
- โ Find the exact value of $\cos(105^{\circ})$
- โ Find the exact value of $\tan(195^{\circ})$
- โ Simplify $\sin(x + \frac{\pi}{2})$