Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=\frac{\pi}{6}, B=\frac{4 \pi}{3}\)

Step-by-Step Solution

Answer

The exact value is \(-\frac{\sqrt{3}}{2}\).

Step 1: Understanding the Problem

We are asked to find the exact value of \(\cos(A - B)\) for the angles \(A = \frac{\pi}{6}\) and \(B = \frac{4\pi}{3}\). The formula for \(\cos(A - B)\) is given by \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). To proceed, we need to find the values of \(\cos A\), \(\cos B\), \(\sin A\), and \(\sin B\).

Step 1: Find trigonometric values for A and B

First, calculate \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) and \(\sin(\frac{\pi}{6}) = \frac{1}{2}\). Then, calculate \(\cos(\frac{4\pi}{3}) = -\frac{1}{2}\) and \(\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}\).

Step 2: Apply the formula for cos(A-B)

Use the values from Step 1 in the formula: \(\cos(A - B) = \cosA \cosB + \sinA \sinB\). Substituting the values gives: \(\cos(A - B) = \left(\frac{\sqrt{3}}{2}\right)\left(-\frac{1}{2}\right) + \left(\frac{1}{2}\right)\left(-\frac{\sqrt{3}}{2}\right)\).

Step 3: Simplify the expression

The expression simplifies to: \(\cos(A - B) = -\frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4}\). Combining like terms, we get \(- \frac{\sqrt{3}}{2}\).

Step 5: Conclusion: Write the final answer

The exact value of \(\cos\left(\frac{\pi}{6} - \frac{4\pi}{3}\right)\) is \(- \frac{\sqrt{3}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry. They allow us to express and manipulate trigonometric functions more flexibly. One important identity used in various applications is the cosine difference identity, which is helpful in calculating the cosine of the difference between two angles, such as \(A - B\).The cosine difference identity is described by the formula:

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

This identity is derived from the more general trigonometric addition formulas, enabling the calculation of cosine for angles not directly found on the unit circle. Understanding these identities helps solve complex trigonometric expressions easily by breaking them down into solvable components. By learning and applying trigonometric identities, students can simplify calculations and deepen their understanding of trigonometric relationships.

Exact Trigonometric Values

Exact trigonometric values are specific values of trigonometric functions at certain angles that can be expressed without decimals. Typical examples include angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), and so on. These angles can be represented on the unit circle, making it easy to recognize their sine and cosine values.For instance, for the angle \(\frac{\pi}{6}\):

• \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
• \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)

And for \(\frac{4\pi}{3}\) (which lies in the third quadrant):

• \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}\)
• \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)

These exact values help simplify calculations, such as finding the exact value of trigonometric expressions, and are often remembered through the unit circle or mnemonic devices.

Angle Subtraction in Trigonometry

Angle subtraction in trigonometry is a method used to determine trigonometric values for expressions like \(\cos(A - B)\). It involves understanding the relationships between two angles and applying identities to find exact values. This concept extends the basic trigonometric functions to more complex scenarios by incorporating both difference and sum identities.When subtracting angles, such as \(A = \frac{\pi}{6}\) and \(B = \frac{4\pi}{3}\), the cosine difference identity is used. By using known exact trigonometric values for each angle, the problem can be systematically solved effortlessly:Determine the values as:

• \( \cos A = \frac{\sqrt{3}}{2}\), \( \sin A = \frac{1}{2}\)
• \( \cos B = -\frac{1}{2}\), \( \sin B = -\frac{\sqrt{3}}{2}\)

Applying the identity, \(\cos(A - B) = \frac{\sqrt{3}}{2} \times -\frac{1}{2} + \frac{1}{2} \times -\frac{\sqrt{3}}{2}\) results in a simplified answer of \(-\frac{\sqrt{3}}{2}\), highlighting the importance of the trigonometric difference identity in practical applications. Understanding angle subtraction helps solve a wide range of trigonometric problems, reinforcing the link between theory and practice.