Problem 12 - Precalculus: Functions and Graphs

Question

Exer. \(11-16\) : Express as a trigonometric function of one angle. $$\cos 13^{\circ} \cos 50^{\circ}-\sin 13^{\circ} \sin 50^{\circ}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \cos 63^{\circ} \).

Step 1: Recognize the Identity

Notice that the expression \( \cos A \cos B - \sin A \sin B \) closely resembles the cosine angle addition formula, which is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).

Step 2: Apply the Cosine Addition Formula

Apply the cosine addition formula to the given expression \( \cos 13^{\circ} \cos 50^{\circ} - \sin 13^{\circ} \sin 50^{\circ} = \cos(13^{\circ} + 50^{\circ}) \).

Step 3: Simplify the Angle

Add the angles: \( 13^{\circ} + 50^{\circ} = 63^{\circ} \). The expression simplifies to \( \cos 63^{\circ} \).

Step 4: Final Expression

The given trigonometric expression \( \cos 13^{\circ} \cos 50^{\circ} - \sin 13^{\circ} \sin 50^{\circ} \) simplifies to \( \cos 63^{\circ} \).

Key Concepts

Cosine Addition Formula

The cosine addition formula is a powerful trigonometric identity that helps you combine two angles into a single trigonometric expression. It's an essential tool especially when working with trigonometric expressions that involve sums or differences of angles.

Mathematically, the formula is expressed as:

• \[\cos(A + B) = \cos A \cos B - \sin A \sin B\]

This formula allows you to simplify trigonometric expressions that appear complex at first glance. This is done by converting them into simpler expressions involving the cosine of a single angle.

In our original exercise, the expression \( \cos 13^{\circ} \cos 50^{\circ} - \sin 13^{\circ} \sin 50^{\circ} \) was identified as matching the cosine addition formula, ultimately simplifying into just \( \cos 63^{\circ} \).

Understanding how to apply this formula will not only help solve trigonometric problems more efficiently but also build a deeper understanding of angle manipulation in trigonometry.

Angle Addition

Angle addition is the concept of combining two angles to form a new angle. It is a key topic in trigonometry, which is often required when applying trigonometric identities such as the cosine addition formula.

When you add angles together, you can express trigonometric functions of these combined angles using known formulas, as seen with the polynomial-like identity used in cosine addition:

Given angles \( A \) and \( B \), the identity becomes:

• \[A + B\]

In our exercise, adding the angles \( 13^{\circ} \) and \( 50^{\circ} \) gives us \( 63^{\circ} \). This process simplifies multi-angle trigonometric expressions into a more manageable form of single angle expressions.

Grasping angle addition is crucial for simplifying expressions and solving problems that involve multiple angle scenarios. It not only aids in the application of trigonometric identities but also enhances problem-solving strategies in broader mathematical contexts.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing complex trigonometric expression into a much simpler and more understandable form. The goal is to make computation easier and results clearer.

A useful method to achieve simplification is through the application of trigonometric identities like the cosine addition formula. These identities serve as shortcuts to reduce the laborious calculations involved in evaluating trigonometric expressions with multiple angles.

Let's consider how simplification works in our exercise. By applying the cosine addition formula:

• Original: \( \cos 13^{\circ} \cos 50^{\circ} - \sin 13^{\circ} \sin 50^{\circ} \)
• Simplified: \( \cos 63^{\circ} \)

This exemplifies how an expression with two angles can be reduced to an expression with only one angle. Simplification does not just apply to textbook problems but is an invaluable skill in various real-world and theoretical math problems, saving time and effort in solving complex equations.