Problem 5 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \cos 105^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 105^{\circ} \) is \( \frac{\sqrt{2} - \sqrt{6}}{4} \).

Step 1: Identify Appropriate Formula

To solve this problem, we can use the cosine addition formula: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \). We need to express \( 105^{\circ} \) as a sum of angles we know the exact values of, such as \( 60^{\circ} + 45^{\circ} \).

Step 2: Apply the Addition Formula

Substitute \( a = 60^{\circ} \) and \( b = 45^{\circ} \) into the addition formula: \( \cos(105^{\circ}) = \cos(60^{\circ} + 45^{\circ}) = \cos 60^{\circ} \cdot \cos 45^{\circ} - \sin 60^{\circ} \cdot \sin 45^{\circ} \).

Step 3: Find Exact Trigonometric Values

Recall the exact trigonometric values: \( \cos 60^{\circ} = \frac{1}{2} \), \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), and \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Substitute these into the expression.

Step 4: Simplify the Expression

Substituting the values into the formula: \( \cos 105^{\circ} = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \).

Step 5: Finalize Result

Combine the fractions: \( \cos 105^{\circ} = \frac{\sqrt{2} - \sqrt{6}}{4} \). This is the exact value of \( \cos 105^{\circ} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that simplify the computation of expressions involving trigonometric functions. They are true for any angle and help in solving equations and finding exact values of angles. Understanding trigonometric identities is crucial for solving complex problems in trigonometry, such as the one we encounter with cosine and sine functions.
Let's break down some common trigonometric identities:

• Pythagorean Identity: This is one of the most well-known, and it states that \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \).
• Angle Sum and Difference Identities: These include formulas for \( \sin(a \pm b) \) and \( \cos(a \pm b) \), which allow us to find the sine and cosine of sums and differences of angles.
• Double Angle Formulas: These formulas, such as \( \sin(2a) = 2 \sin a \cos a \) and \( \cos(2a) = \cos^2 a - \sin^2 a \), assist in working with angles that are multiples of other angles.

By understanding and applying these identities, you can solve problems involving angles that aren't immediately obvious or easily evaluated with simple trigonometric ratios.

Exact Trigonometric Values

Exact trigonometric values are values of the trigonometric functions that can be calculated exactly, rather than approximately. These values are based on angles that appear frequently in trigonometry, such as \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \). Knowing these values by heart allows for quick evaluation of more complex expressions.
Some important exact values are:

• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

Using these exact values, you can simplify trigonometric expressions. For example, when calculating the cosine of \( 105^{\circ} \), which can be expressed as \( 60^{\circ} + 45^{\circ} \), applying these values is pivotal.

Cosine Addition Formula

The cosine addition formula is a valuable identity that allows you to calculate the cosine of an angle resulting from the sum of two separate angles. This formula is given by: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \). It shows how the cosine of the sum of two angles relates to the cosine and sine of each angle separately.
In practical use, this formula breaks down into these steps:

• Identify angles \( a \) and \( b \) whose sum is the angle you wish to find.
• Compute \( \cos a \), \( \cos b \), \( \sin a \), and \( \sin b \) using known exact values or approximations.
• Substitute these values into the formula \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \).
• Simplify the result to find the exact value.

With this formula, finding the cosine of angles like \( 105^{\circ} \), by breaking it down into \( 60^{\circ} + 45^{\circ} \), becomes straightforward. Properly applying this formula will yield an exact trigonometric value without needing a calculator.