Problem 128 - Calculus Volume 1

Question

Evaluate the following functional values. $$\cos \left(\frac{5 \pi}{12}\right)$$

Step-by-Step Solution

Answer

\(\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2} - \sqrt{6}}{4}\)

Step 1: Express the Angle as a Sum of Known Angles

To evaluate the cosine of \(\frac{5\pi}{12}\), we need to express the angle as a sum or difference of angles for which we know the exact trigonometric values. Let's rewrite \(\frac{5\pi}{12}\) as \(\frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}\).

Step 2: Use the Cosine Sum Formula

The cosine of a sum of two angles \(a\) and \(b\) is calculated using the formula: \(\cos(a+b) = \cos a \cos b - \sin a \sin b\). Here, \(a = \frac{\pi}{3}\) and \(b = \frac{\pi}{4}\).

Step 3: Substitute the Known Values of Trigonometric Functions

Use the known values for cosine and sine: - \(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\)- \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)- \(\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)- \(\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Step 4: Plug the Values into the Cosine Sum Formula

Substitute the values into the cosine sum formula:\[ \cos \left(\frac{5\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \]

Step 5: Simplify the Expression

Simplify the expression:\[ \cos \left(\frac{5\pi}{12}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \] Combine into a single fraction: \[ \cos \left(\frac{5\pi}{12}\right) = \frac{\sqrt{2} - \sqrt{6}}{4} \]

Step 6: Write the Final Answer

The cosine of \(\frac{5\pi}{12}\) is \(\frac{\sqrt{2} - \sqrt{6}}{4}\).

Key Concepts

Cosine Sum Formula

The cosine sum formula is a key tool in trigonometry for evaluating the cosine of the sum of two angles. It is expressed as:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

This formula helps in cases where the angle is not one of the basic angles. Instead, the sum formula uses known angles like \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) to find more complex values.

For instance, evaluating \( \cos \left( \frac{5\pi}{12} \right) \) involves breaking it down to manageable parts: \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \), then applying the cosine sum formula to find an exact value. This approach is powerful for precise calculations, especially when calculators or numerical approximations are not allowed.

Angle Sum Identity

The angle sum identity in trigonometry provides a framework for expressing the trigonometric functions of an angle in terms of the angles that make up its sum. Specifically here, it's used for the cosine function. By writing \( \frac{5\pi}{12} \) as \( \frac{\pi}{3} + \frac{\pi}{4} \), we can apply the angle sum identity directly.

This identity is vital since it allows us to use more straightforward angles for which we have exact trigonometric values. Utilizing these known angles, computation becomes an algebraic process rather than complex estimation. The angle sum identity thus broadens our toolkit, making it easier to solve challenging trigonometric problems by breaking them down into simpler components.

Exact Trigonometric Values

To effectively use the cosine sum formula and the angle sum identity, it's crucial to know the exact trigonometric values of standard angles. These include angles like \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), from which we can derive precise values for sine and cosine:

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

These fundamental values are crucial because they are readily memorizable, providing a fast and accurate way to compute exact values for more complex angles, making trigonometry less intimidating and more accessible for students.