Problem 16 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. $$\cos 10^{\circ} \cos 80^{\circ}-\sin 10^{\circ} \sin 80^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the expression is 0.

Step 1: Identify Trigonometric Identity

The expression \( \cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ} \) is similar to the cosine addition formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Here, the expression matches this identity.

Step 2: Substitute Values into the Formula

Identify \( a = 10^{\circ} \) and \( b = 80^{\circ} \). Substitute these values into the cosine addition formula: \( \cos(10^{\circ} + 80^{\circ}) = \cos 90^{\circ} \).

Step 3: Evaluate the Trigonometric Function

The expression simplifies to \( \cos 90^{\circ} \). We know that \( \cos 90^{\circ} = 0 \). Therefore, the exact value of the original expression is 0.

Key Concepts

Cosine addition formula

The cosine addition formula is a crucial tool in trigonometry. It allows you to express trigonometric functions as a single function by combining angles. This identity is written as:

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

This formula is helpful when working with angles that are difficult to evaluate on their own. The formula helps convert complex expressions into simpler, standardized forms. In the context of our exercise, we used it to transform \( \cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ} \) into \( \cos 90^{\circ} \).
This transformed the problem into something easily solved using known trigonometric values.
When applying the formula, it is essential to identify the values for \(a\) and \(b\) correctly and ensure that they add up to the desired angle. Once computed, you can quickly find the solution to the expression.

Exact value

In trigonometry, the term "exact value" refers to the precise value of a trigonometric function at a specific angle, without approximation. These values are fundamental, as they form the basis for trigonometric calculations in geometry and physics.
An exact value is often determined using known properties of key angles within the unit circle, such as \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \text{and} 90^{\circ}\). In this exercise, after applying the cosine addition formula, we are left with the expression \( \cos 90^{\circ} \).

The exact value of \( \cos 90^{\circ} \) is \(0\), because at \(90^{\circ}\), the cosine function, which measures the horizontal distance from the origin on the unit circle, is 0. Knowing these exact trigonometric values is critical, as it provides a foundation for resolving more complex problems without resorting to numerical approximations.

Trigonometric functions

Trigonometric functions include sine, cosine, and tangent, among others. They are functions based on the ratios of the sides of a right triangle relative to its angles. These functions are crucial in understanding the relationships within geometric figures and modeling periodic phenomena such as waves.

Each trigonometric function has an associated graph and set of values within the unit circle. For example:

• Sine (\(\sin\)) - represents the vertical component in the unit circle.
• Cosine (\(\cos\)) - represents the horizontal component in the unit circle.
• Tangent (\(\tan\)) - the ratio of sine to cosine.

When working with these functions, it’s common to use identities like the cosine addition formula to simplify expressions. Understanding how to rewrite combinations of angles and values with these identities forms the basis for performing more complex calculations under trigonometry. Recognizing how to derive exact values of trigonometric functions and utilize these identities is a key skill in mathematics.