Problem 108 - 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 \(t\) $$t=-\pi$$

Step-by-Step Solution

Answer

The exact value of \( \cos 3t \) for \( t=-\pi \) is -1 and the exact value of \( \cos \left(\frac{t}{3}\right) \) for \( t=-\pi \) is \(\frac{1}{2}\)

Step 1: Substitute the given value of t in \(\cos 3t\)

Substitute \( t = -\pi \) into the expression \( \cos 3t \). So, we have \( \cos (3(-\pi)) \) which simplifies to \( \cos(-3\pi) \).

Step 2: Find the value of \(\cos(-3\pi)\)

The function \( \cos \) has a period of \(2\pi\), therefore, \( \cos(-3\pi) = \cos(\pi) \) because \( -3\pi \) and \( \pi \) are coterminal angles. The cosine of \( \pi \) is \( -1 \). So, \( \cos(-3\pi) = -1\) .

Step 3: Substitute the given value of t in \(\cos\left(\frac{t}{3}\right)\)

Substitute \( t = -\pi \) into the expression \( \cos \left(\frac{t}{3}\right) \). So, we have \( \cos \left(\frac{-\pi}{3}\right) \).

Step 4: Find the value of \(\cos \left(\frac{-\pi}{3}\right)\)

\(\cos \left(\frac{-\pi}{3}\right) \) is negative in the third and fourth quadrants. Since \( \frac{-\pi}{3} \) is a negative angle and lies in the fourth quadrant, the result would be positive and amount to \( \cos \left(\frac{\pi}{3}\right)=\frac{1}{2}\) because in the fourth quadrant cosine is positive.

Key Concepts

Cosine

Cosine is one of the primary trigonometric functions, which relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. For an angle \( \theta \), this is written as \( \cos(\theta) \). Cosine values repeat periodically as angles increase or decrease, making it versatile for various applications.

Here's what to remember about cosine:

• Range: Cosine values lie between \(-1\) and \(1\).
• Period: The function repeats itself every \(2\pi\).
• Key angles: Familiarize yourself with common angles like \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).

Understanding these will greatly help in evaluating expressions like \( \cos 3t \) or \( \cos \left(\frac{t}{3}\right) \).

Angle Transformation

Angle transformation involves modifying an angle, often to make it easier to work with. This technique can help simplify trigonometric expressions, as with \( \cos(3t) \) where the angle \( t \) is multiplied by 3.

When transforming angles, you typically:

• Change the multiplication factor, like \( 3t \) becoming \( -3\pi \).
• Switch between negative and positive angles, using their properties to find cosine or other trigonometric values.

Practicing these transformations enhances understanding of how angles adjust within trigonometric contexts.

Coterminal Angles

Coterminal angles are angles in standard position that share the same terminal side, meaning they end up in the same place on the unit circle. For instance, angles like \(-3\pi\) and \(\pi\) are coterminal.

How do you find coterminal angles?

• Add or subtract \(2\pi\) repeatedly to reach an equivalent angle.

Understanding coterminal angles is crucial, especially in solving expressions like \( \cos(-3\pi) \). This concept shows how angles can look different but have the same trigonometric values due to their positions.

Periodicity of Trigonometric Functions

Periodicity in trigonometry refers to the repeating nature of functions like cosine. For cosine, this period is \(2\pi\), meaning the function's values recur every \(2\pi\) radians.

Key points about periodicity:

• Cosine will always reach the same value once a full cycle (\(2\pi\)) is completed.
• This property helps simplify complex calculations, like converting \(-3\pi\) to \(\pi\).

Grasping periodicity is essential as it enables predicting function behavior over large intervals, reaffirming the consistency of trigonometric functions.