Problem 105 - 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 \(t\) $$t=0$$

Step-by-Step Solution

Answer

\(\cos 3 t = 1\) and \(\cos \left(\frac{t}{3}\right) = 1\) for \(t=0\).

Step 1: Identify and Substitute

The variable \(t\) is given as 0. Substitute \(t=0\) into \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\). This results in \(\cos(0)\) in both cases.

Step 2: Calculate Cosine values

The cosine of 0 is 1, so we find \(\cos(0) = 1\).

Key Concepts

Cosine

Cosine is one of the primary trigonometric functions, and it is essential in understanding the relationship between angles and lengths in right triangles. In the context of the unit circle, the cosine of an angle gives the x-coordinate of the point where the terminal side of the angle intersects the circle. This is why it is often described as measuring "the distance along the x-axis."

Some key characteristics of the cosine function include:

• Periodicity: Cosine is a periodic function with a period of \(2\pi\), meaning that \(\cos(\theta + 2\pi) = \cos(\theta)\) for any angle \(\theta\).
• Range: The values of the cosine function lie between -1 and 1, inclusive.
• Evenness: Cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\). This property makes it symmetrical about the y-axis.

The cosine function is particularly useful for calculating the length of an adjacent side in right-angled triangles when a hypotenuse is known. By understanding cosine, students can apply it to a variety of problems involving angles and circles.

Angles

Understanding angles is fundamental in trigonometry as angles are used to define the trigonometric functions such as sine, cosine, and tangent. An angle is formed by two rays with a common starting point called the vertex. In trigonometry, angles can be measured in degrees or radians.

Radians offer a natural way to measure angles in mathematics, where a full circle is \(2\pi\) radians around, as opposed to 360 degrees in the degree measure. Each angle measurement has its advantages, with radians often used in higher mathematics because of their natural link to calculus.

When dealing with trigonometric functions:

• It is important to be consistent with angle measurement units (degrees vs radians).
• Some angles have well-known trigonometric values, such as 0, \(\frac{\pi}{2}\), \(\pi\), and \(2\pi\).
• For example, \(\cos(0) = 1\), which matches our exercise.

Angled problems often involve translating between radians and degrees and recognizing these special angle values to easily work with trigonometric expressions and functions.

Exact Values

In the realm of trigonometry, calculating exact values of trigonometric functions like cosine is a crucial skill. Exact values refer to non-decimalized outputs that are typically expressed in terms of square roots or integers rather than decimals. This is essential for precise calculations and many advanced mathematical problems.

For commonly used angles, such as \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and their multiples, the exact values of trigonometric functions are well-established:

• \(\cos(0) = 1\)
• \(\cos(\frac{\pi}{3}) = \frac{1}{2}\)
• \(\cos(\frac{\pi}{2}) = 0\)
• \(\cos(\pi) = -1\)

These values are derived from the properties of the unit circle and can be memorized because of their frequent use. Understanding these values helps solve trigonometric problems more quickly and efficiently without relying on a calculator for approximation.