Problem 34 - Precalculus with Calculus Previews

Question

Use a sum-to-product-formula in Theorem 4.7.2 to find the exact value of the expression. Do not use a calculator. $$ \cos 15^{\circ}+\cos 75^{\circ} $$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{6}}{2} \).

Step 1: Recall the Sum-to-Product Formula

The sum-to-product formula for cosine is given by \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \). We will use this formula to simplify \( \cos 15^{\circ} + \cos 75^{\circ} \).

Step 2: Identify A and B

In this exercise, let \( A = 75^{\circ} \) and \( B = 15^{\circ} \). We need to find \( A + B \) and \( A - B \).

Step 3: Find A + B and A - B

Calculate \( A + B = 75^{\circ} + 15^{\circ} = 90^{\circ} \) and \( A - B = 75^{\circ} - 15^{\circ} = 60^{\circ} \).

Step 4: Apply the Formula

Using the sum-to-product formula, substitute these results: \[ \cos 15^{\circ} + \cos 75^{\circ} = 2 \cos \left( \frac{90^{\circ}}{2} \right) \cos \left( \frac{60^{\circ}}{2} \right). \]

Step 5: Simplify the Formula

Compute the individual cosines: \( \cos \left( \frac{90^{\circ}}{2} \right) = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) and \( \cos \left( \frac{60^{\circ}}{2} \right) = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 6: Calculate the Product

Substitute the simplified terms back into the formula: \[ 2 \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) = \frac{\sqrt{6}}{2}. \]

Step 7: Finalize the Calculation

Conclude that the exact value of \( \cos 15^{\circ} + \cos 75^{\circ} \) is \( \frac{\sqrt{6}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in the field of mathematics. They are equations involving trigonometric functions that are true for all values of the variable where both sides of the equation are defined. These identities help simplify complex trigonometric expressions, making it easier to evaluate them or solve equations.
Some of the most common trigonometric identities include Pythagorean identities, angle sum and difference identities, and product-to-sum and sum-to-product identities.
The sum-to-product identities, specifically, allow us to express sums of trigonometric functions like sine and cosine as products. In our specific exercise, the sum-to-product formula for cosine: \[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \] was used to simplify and find the exact value of the expression \( \cos 15^{\circ} + \cos 75^{\circ} \).
Utilizing these identities improves efficiency and simplifies calculations without the use of calculators. This process shows the power of mathematical relationships.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent play a crucial role in understanding angles and lengths in different geometrical contexts and applications. The cosine function, which was our primary focus in this exercise, relates the adjacent side of a right triangle to its hypotenuse. It's denoted as \( \cos \Theta \).
For any angle \( \Theta \), \( \cos \Theta \) takes a value ranging from -1 to 1.

• Cosine of an angle in a right triangle can be given by the ratio of the length of the adjacent side to the hypotenuse.
• In the unit circle framework, cosine represents the horizontal coordinate of the point corresponding to angle \( \Theta \).

These trigonometric functions are periodic and have specific values at notable angles such as \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \, \text{and} \, 90^{\circ} \).
Therefore, by understanding these, we could calculate the value of \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) directly from fundamental trigonometric tables without computation, aiding us in completing the exercise.

Exact Values of Trigonometric Expressions

Finding the exact values of trigonometric expressions without a calculator requires a solid understanding of trigonometric identities and functions. The precise calculation of such expressions relies on specific known values and identities that can simplify or break down the components of an expression.
In the exercise, we found the exact value of \( \cos 15^{\circ} + \cos 75^{\circ} \) using the sum-to-product formula. Here, each transformation step leveraged well-known trigonometric values.

• We identified known angles such as \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \text{and} \, 90^{\circ} \), whose exact trigonometric values are memorably simple (like \( \frac{\sqrt{2}}{2} \) or \( \frac{\sqrt{3}}{2} \)).
• These exact values, derived from geometric principles, remain crucial for tackling such problems.

Through using these values and applying the sum-to-product formula, we ultimately solved \( \cos 15^{\circ} + \cos 75^{\circ} \) accurately, showing the power of understanding exact trigonometric values in arithmetic manipulations.