Pre-Calculus: Definition and Explanation of Sum and Difference Identities

📚 Understanding Sum and Difference Identities

Sum and difference identities are trigonometric identities that allow you to find the trigonometric values of angles that are the sum or difference of two other angles. They're super useful when you don't have a calculator handy or need to find exact values.

📐 The Formulas

Here's a breakdown of the formulas you'll need:

• ➕ Sine of a Sum: $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• ➖ Sine of a Difference: $\sin(A - B) = \sin A \cos B - \cos A \sin B$
• ➕ Cosine of a Sum: $\cos(A + B) = \cos A \cos B - \sin A \sin B$
• ➖ Cosine of a Difference: $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• ➕ Tangent of a Sum: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
• ➖ Tangent of a Difference: $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

📝 Let's Break it Down

Each identity expresses the trigonometric function (sine, cosine, or tangent) of a sum or difference of two angles (A and B) in terms of the trigonometric functions of the individual angles A and B.

💡 Example Time!

Let's find the exact value of $\sin(75^\circ)$. We can rewrite $75^\circ$ as $45^\circ + 30^\circ$. Now, we can use the sine of a sum identity:

$\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)$

We know: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, $\sin(30^\circ) = \frac{1}{2}$, and $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.

Plugging these values in, we get:

$\sin(75^\circ) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = \frac{\sqrt{6} + \sqrt{2}}{4}$

So, $\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$

🍎 Teacher's Guide: Sum and Difference Identities

Objectives:

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• Students will be able to state the sum and difference identities for sine, cosine, and tangent.
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• Students will be able to apply the sum and difference identities to evaluate trigonometric functions of non-standard angles.
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• Students will be able to simplify trigonometric expressions using sum and difference identities.

Materials:

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• Whiteboard or projector
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• Markers or pens
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• Handout with the formulas
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• Practice problems with answer key

Warm-up (5 minutes):

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• Review the unit circle and trigonometric values for special angles ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$).
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• Ask students to recall $\sin(30^\circ)$, $\cos(45^\circ)$, etc.

Main Instruction (30 minutes):

1. Introduce the sum and difference identities. Write each identity on the board.
2. Explain what each variable represents and why these identities are useful.
3. Work through several examples. Start with simple examples and gradually increase the difficulty.
4. Emphasize the importance of memorizing or having the identities readily available.

Assessment (10 minutes):

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• Give students a short quiz with problems that require the use of sum and difference identities.
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• Collect and grade the quizzes to assess student understanding.

🎯 Practice Quiz

Solve the following problems using sum and difference identities:

1. Find the exact value of $\cos(15^\circ)$.
2. Find the exact value of $\tan(105^\circ)$.
3. Simplify the expression: $\sin(x + \frac{\pi}{2})$.
4. Simplify the expression: $\cos(x - \frac{\pi}{2})$.
5. Find the exact value of $\sin(165^\circ)$.
6. Find the exact value of $\cos(345^\circ)$.
7. Simplify the expression: $\tan(x + \pi)$.