Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=\frac{\pi}{2}, B=\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(A+B) \) is \(-\frac{\sqrt{3}}{2}\).

Step 1: Identify the Formula

To find \( \cos(A+B) \), we use the cosine addition formula: \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] This formula will allow us to calculate \( \cos(A+B) \) using the values of \( A \) and \( B \).

Step 2: Define Known Values

Given \( A = \frac{\pi}{2} \) and \( B = \frac{2\pi}{3} \), we need to find \( \cos A, \cos B, \sin A, \) and \( \sin B \).

Step 3: Calculate \( \cos A \) and \( \sin A \)

Using unit circle values, \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \).

Step 4: Calculate \( \cos B \) and \( \sin B \)

Using unit circle values, \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \) and \( \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \).

Step 5: Substitute Values into the Formula

Substitute the known values into the cosine addition formula: \[ \cos(A+B) = (0)(-\frac{1}{2}) - (1)(\frac{\sqrt{3}}{2}) \].

Step 6: Calculate the Result

Perform the multiplications and additions: \( \cos(A+B) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle Values

Understanding unit circle values is crucial in trigonometry as they provide the exact values for sine and cosine at various angles. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It's a helpful tool for visualizing trigonometric functions.

Key angles on the unit circle include multiples of \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},\) and \(\frac{\pi}{2}\).
These angles, along with their sine and cosine values, are widely used because they provide exact trigonometric values which are easy to remember and apply.

• At \(\frac{\pi}{2}\), the coordinates are \((0,1)\), meaning \(\cos(\frac{\pi}{2})=0\) and \(\sin(\frac{\pi}{2})=1\).
• At \(\frac{2\pi}{3}\), reflection over the y-axis gives you coordinates of \((-\frac{1}{2}, \frac{\sqrt{3}}{2})\), so \(\cos(\frac{2\pi}{3})=-\frac{1}{2}\) and \(\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}\).

Grasping these values and their locations on the unit circle helps simplify solving trigonometric equations, such as using the cosine addition formula.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variable.
They are foundational tools for simplifying and solving trigonometry-related problems, especially in calculating angles and lengths in various shapes.

Among the most essential identities are the Pythagorean identities, angle sum and difference identities, and double angle formulas. Let's focus on the cosine addition formula:

• The cosine addition formula states: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
• It expresses \(\cos(A+B)\) in terms of the cosine and sine of \(A\) and \(B\).

Using this identity allows us to find the cosine of the sum of two angles when their individual sine and cosine values are known. In real-world applications, these identities enable calculations such as finding unknown side lengths in triangles or analyzing wave functions.

Exact Trigonometric Values

Exact trigonometric values are non-decimal representations of trigonometric function outputs for specific angle measures.
These values are typically preferred over decimal approximations as they provide precise results without rounding errors.

Common angles on the unit circle, like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},\) and \(\frac{\pi}{2}\), have exact values that are often simple fractions or involve \(\sqrt{2}\) or \(\sqrt{3}\).

• The trigonometric values of these angles, such as \(\sin(\frac{\pi}{2})=1\) or \(\cos(\frac{\pi}{3})=\frac{1}{2}\), aid in diverse calculations without the need for approximation.
• Since these values are derived from exact equations, they help maintain accuracy and consistency in mathematical solutions.

As seen in the exercise, using exact values for \(\cos A\), \(\sin A\), \(\cos B\), and \(\sin B\), simplifies the calculation process and helps achieve a precise final result without computational errors.