Problem 3 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin 75^{\circ} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Two Angles

Determine two angles whose sum equals the given angle, 75°. We can use 45° and 30° since their sum is 75°.

Step 2: Recall the Addition Formula

The formula for the sine of the sum of two angles is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 3: Apply the Formula

Apply the addition formula for \( \sin(75^{\circ}) = \sin(45^{\circ} + 30^{\circ}) \), which gives: \( \sin(75^{\circ}) = \sin(45^{\circ}) \cos(30^{\circ}) + \cos(45^{\circ}) \sin(30^{\circ}) \).

Step 4: Substitute Exact Values

Use the exact trigonometric values: \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \), and \( \sin(30^{\circ}) = \frac{1}{2} \).

Step 5: Calculate

Substitute these values into the formula: \( \sin(75^{\circ}) = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) + \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) \). Simplify to: \( \sin(75^{\circ}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \).

Step 6: Simplify Further

Combine the fractions: \( \sin(75^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4} \). This is the exact value of the expression.

Key Concepts

Addition Formula

Trigonometric identities often involve formulas like the addition formula which are essential in solving problems involving specific angles. The addition formula is particularly useful because it allows us to find the sine or cosine of a sum of two angles. In many exercises, our input angle isn't directly in the list of standard angles that we know the exact values for, such as 30°, 45°, or 60°.

The sine addition formula is given by:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

To use this, select two angles, say \(a\) and \(b\), whose sum gives the desired angle. These angles should ideally be those for which you know the exact trigonometric values. For example, when finding \(\sin(75^{\circ})\), we choose \(45^{\circ}\) and \(30^{\circ}\), since their sum equals 75° and both have simple exact values.

Exact Trigonometric Values

To solve trigonometric problems involving specific angles, having a good grasp of exact trigonometric values is crucial. These are known values for sine, cosine, and other functions at significant angles. Such values often appear in many calculations, allowing you to simplify expressions without a calculator.

Here are some exact values commonly used:

• \( \sin(30^{\circ}) = \frac{1}{2} \)
• \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)
• \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)
• \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)

These exact values are essential when applying the addition and angle sum identities, allowing you to substitute these values directly into the formula and simplify the expression efficiently. Remembering these exact values can save you time and avoid common calculation errors.

Angle Sum Identities

Angle sum identities are critical when solving problems involving angles that are built from simpler, well-known angles. They provide a way to break down complex trigonometric functions into more manageable parts. This is especially helpful in both scientific fields and practical applications like engineering and architecture.

The key angle sum identities you should remember include:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

Using these identities, you can express many angles as sums or differences of known angles. If you need to find \(\sin(75^{\circ})\), it's easier to rewrite it as \(\sin(45^{\circ} + 30^{\circ})\) because you utilize known values for \(\sin\) and \(\cos\) at these component angles. Mastery of these identities can greatly simplify complex trigonometric problems.