Problem 106 - Precalculus : Building Concepts and Connections

Question

Find exact values of \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\) for the given values of \(t\) $$t=\frac{\pi}{2}$$

Step-by-Step Solution

Answer

The exact values for \(\cos 3 t\) is \(0\) and \(\cos \left(\frac{t}{3}\right)\) is \(\frac{\sqrt{3}}{2}\) when \(t = \frac{\pi}{2}\).

Step 1: Apply the formula for cosine of 3t

We know that \(\cos 3 t = 4\cos^3 t - 3\cos t\). We substitue \(t = \frac{\pi}{2}\) into the equation. This simplifies to \(\cos 3 t = 4\cos^3(\frac{\pi}{2}) - 3\cos(\frac{\pi}{2})\).

Step 2: Compute value of cos(3t)

We know that \(\cos(\frac{\pi}{2}) = 0\). Then, \(\cos 3 t = 4*0^3 - 3*0 = 0\).

Step 3: Determine value of cos(t/3)

However, we need to be careful when determining \(\cos(\frac{t}{3})\), as division by 3 makes only sense when the result still falls within the standard range for arguments of the cosine function, i.e., \([- \pi, \pi]\). When \(t = \frac{\pi}{2}\), we get \(\frac{t}{3} = \frac{\pi}{6}\). So, \(\cos\left(\frac{t}{3}\right) = \cos\left(\frac{\pi}{6}\right)\).

Step 4: Compute value of cos(t/3)

We know that \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\). Thus, the exact value for \(\cos\left(\frac{t}{3}\right)\) when \(t = \frac{\pi}{2}\) is \(\frac{\sqrt{3}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all inputs of angles. They are incredibly useful tools for solving complex trigonometric problems, simplifying expressions, and proving mathematical theorems. These identities work as shortcuts, saving time and effort in computations. Some of the most common identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle sum and difference identities, like \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
• Double and half angle identities, for example, \( \cos 2\theta = 2\cos^2 \theta - 1 \).

Understanding these identities is crucial, as they are applied to solve exercises like finding \( \cos 3t \) and \( \cos \frac{t}{3} \), leading to simplification and precise computation.

Cosine Triple Angle Formula

The cosine triple angle formula expresses \( \cos 3t \) in terms of \( \cos t \). The formula is given by:\[\cos 3t = 4\cos^3 t - 3\cos t\]This formula is derived from the angle addition identities and powers of trigonometric functions. By expressing \( \cos 3t \) via this formula, we can calculate the cosine of three times an angle if we know the cosine of the base angle, \( t \). In our solution to find \( \cos 3t \) when \( t = \pi/2 \), we substitute \( t = \pi/2 \) into this formula, leading to simplification:- Since \( \cos(\pi/2) = 0 \), the equation becomes \[\cos 3t = 4 \times 0^3 - 3 \times 0 = 0\]This example demonstrates how utilizing the cosine triple angle formula can streamline finding exact trigonometric values, especially when direct computation of \( \cos 3t \) is complex or cumbersome.

Angle Reduction

Angle reduction simplifies the calculation of trigonometric functions by reducing the measure of the angle to a more manageable size. It usually involves dividing or modifying the angle so that it fits within the key interval of \([-\pi, \pi]\) or \([0, 2\pi]\).These smaller angle values often correspond to well-known standard angles, allowing us to leverage known exact trigonometric values. In the problem given, we need to compute \( \cos(\frac{t}{3}) \) when \( t = \frac{\pi}{2} \). By utilizing angle reduction, we get: \[\frac{t}{3} = \frac{\pi}{6}\]The familiar angle of \( \frac{\pi}{6} \) falls into the commonly used trigonometric values, giving us: \[\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\]Effective angle reduction not only ensures simplicity and accuracy but also highlights the use of special angle values that streamline solving trigonometric problems.