Problem 17 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\cos(A+B) = \frac{\sqrt{3} - 1}{2\sqrt{2}}\).

Step 1: Identify the Trigonometric Formula

To find \( \cos(A + B) \), we use the cosine addition formula:\[ \cos(A + B) = \cos A \cos B - \sin A \sin B.\]

Step 2: Compute \( \cos A \) and \( \sin A \)

For \( A = \frac{\pi}{4} \), we have \( \cos A = \frac{1}{\sqrt{2}} \) and \( \sin A = \frac{1}{\sqrt{2}} \).

Step 3: Compute \( \cos B \) and \( \sin B \)

For \( B = \frac{\pi}{6} \), the values are \( \cos B = \frac{\sqrt{3}}{2} \) and \( \sin B = \frac{1}{2} \).

Step 4: Substitute Values into the Formula

Substitute the values of \( \cos A, \cos B, \sin A, \) and \( \sin B \) into the formula:\[ \cos(A + B) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2}.\]

Step 5: Simplify the Expression

Calculate each term: \[ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2\sqrt{2}} \] and \[ \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{1}{2\sqrt{2}}.\] Subtract the second term from the first:\[ \cos(A + B) = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}}.\]

Step 6: Combine Like Terms

Combine the terms over a common denominator: \[ \cos(A + B) = \frac{\sqrt{3} - 1}{2\sqrt{2}}.\]

Key Concepts

Cosine Addition Formula

To better understand how to solve the problem of finding the exact value of \(\cos(A + B)\), the Cosine Addition Formula is indispensable. This formula is a key element in trigonometry, making it possible to calculate the cosine of the sum of two angles. The formula is given by:\[\cos(A + B) = \cos A \cos B - \sin A \sin B.\]
This equation allows us to break down complex trigonometric problems into simpler, more manageable calculations. It utilizes smaller trigonometric values of individual angles—\(A\) and \(B\)—to find the cosine of their sum. By computing each component separately, this systematic approach allows for precise evaluations, which are especially useful in a variety of real-world applications, such as physics and engineering.

Understanding this formula is crucial, as it lays the foundation for solving more complex problems involving trigonometric identities. It's a powerful tool in the toolkit of anyone studying trigonometry.

Exact Values of Trigonometric Functions

The exact values of trigonometric functions at specific angles are fundamental in simplifying expressions and solving equations. For angles \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\), knowing these values allows for the straightforward calculation of trigonometric expressions using the addition formula efficiently.

• For an angle of \(\frac{\pi}{4}\):
  • \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\)
  • \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\)
• For an angle of \(\frac{\pi}{6}\):
  • \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)
  • \(\sin \frac{\pi}{6} = \frac{1}{2}\)

Recognizing these exact values is beneficial not just for academic exercises, but also for understanding real-world phenomena where trigonometric functions play a role. Memorizing these can greatly speed up problem-solving and improve your ability to perform mental checks during calculations.

Radians in Trigonometry

Understanding radians is a vital part of trigonometry, as many trigonometric problems use radian measure rather than degrees. Radians provide a natural way of describing angles based on the arc length and the radius of a circle. This is especially advantageous when dealing with mathematical concepts like angular velocity, wave equations, and calculus.

• A radian is defined as the angle formed when the arc of a circle is equal in length to the radius of that circle.
• There are \(2\pi\) radians in a full circle, equivalent to 360 degrees.
• This means that \(\frac{\pi}{4}\) radians converts to 45 degrees, while \(\frac{\pi}{6}\) radians is equivalent to 30 degrees.

Using radian measures allows for a more uniform and integrated approach to mathematics, especially when you encounter higher mathematics, as functions like sine and cosine have derivatives that are simpler with radians than with degrees.
By understanding radians, you can seamlessly transition between different mathematical contexts, enhancing your comprehension and application of trigonometric concepts.