Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=270^{\circ}, B=30^{\circ}\)

Step-by-Step Solution

Answer

The exact value is \(-\frac{1}{2}\).

Step 1: Understand the Problem

We are given angles \(A = 270^{\circ}\) and \(B = 30^{\circ}\) and need to find \(\cos(A - B)\). The formula to use is for the cosine of the difference of two angles: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).

Step 2: Use the Identity for Difference of Angles

According to the formula for cosine of the difference of two angles, we need to find \(\cos A\), \(\cos B\), \(\sin A\), and \(\sin B\). Then apply \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).

Step 3: Find Trigonometric Values

- \(\cos 270^{\circ} = 0\)- \(\sin 270^{\circ} = -1\)- \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)- \(\sin 30^{\circ} = \frac{1}{2}\).

Step 4: Substitute and Calculate

Substitute these values into the identity: \[\cos(270^{\circ} - 30^{\circ}) = \cos 270^{\circ} \cos 30^{\circ} + \sin 270^{\circ} \sin 30^{\circ} \]\[= 0 \times \frac{\sqrt{3}}{2} + (-1) \times \frac{1}{2}\]\[= 0 - \frac{1}{2}\]\[= -\frac{1}{2}\].

Step 5: Final Result

The value of \(\cos(270^{\circ} - 30^{\circ})\) is \(-\frac{1}{2}\).

Key Concepts

Understanding the Cosine Difference Identity

The cosine of the difference of two angles, often referred to as the cosine difference identity, is a fundamental trigonometric identity. It is given by the formula:

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

Here, "A" and "B" are angles. The expression explains how to compute the cosine of the difference between two angles (\( A \) and \( B \)) by knowing the cosine and sine of each angle individually. This identity is incredibly useful because it allows us to break down complex angle differences into simpler, more manageable parts.
Understanding this identity involves not just memorizing the formula but also recognizing how it can apply in various contexts, such as solving equations or finding exact trigonometric values. Its utility lies in its ability to relate the cosine of a complex angle difference back to basic trigonometric functions.

Exact Values of Trigonometric Functions

In trigonometry, some angles have trigonometric values that we can easily determine without a calculator. These are known as exact values. For example, commonly used angles include \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(90^{\circ}\), among others, and their trigonometric functions have set values that are derived from geometric properties or the unit circle.

• For \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• For \( \sin 30^{\circ} = \frac{1}{2} \)
• For \( \cos 270^{\circ} = 0 \)
• For \( \sin 270^{\circ} = -1 \)

These values are crucial for solving trigonometric problems efficiently. By knowing these exact values, we can substitute directly into trigonometric identities like the cosine difference identity, making calculations more straightforward and quicker.

Angle Subtraction in Trigonometry

The concept of angle subtraction is essential when dealing with multiple trigonometric identities. When we talk about subtracting angles in trigonometry, we are referring to the concept of taking one angle and removing the measure of another to find their difference.In the context of trigonometric calculations, knowing how to subtract angles allows us to apply identities like the cosine difference identity. For instance, with angles \( A = 270^{\circ} \) and \( B = 30^{\circ} \), subtracting them gives:

• \( A - B = 270^{\circ} - 30^{\circ} = 240^{\circ} \)

Once we have the resultant angle, we can proceed to determine the trigonometric values or apply further identities as necessary. Understanding angle subtraction is a foundational step in trigonometric problem-solving that enables us to simplify and solve a wide range of trigonometric expressions and equations.