Problem 69 - Algebra and Trigonometry

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$ \text { If } f(\theta)=2 \cos \theta-\cos 2 \theta, \text { find } f\left(\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The exact value of the function \(f(\theta) = 2\cos{\theta} - \cos{2\theta}\) with \(\theta = \frac{\pi}{6}\) is \(f(\frac{\pi}{6}) = \frac{2\sqrt{3} - 1}{2}\).

Step 1: Substitute the Value for θ

First, substitute \(\frac{\pi}{6}\) for \(\theta\) in the function. So, \(f(\frac{\pi}{6}) = 2\cos{\frac{\pi}{6}} - \cos{2\left(\frac{\pi}{6}\right)}\).

Step 2: Evaluate Cosine Fractions

Next, recall that \(\cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2}\). Thus the function becomes \(f(\frac{\pi}{6}) = 2*\frac{\sqrt{3}}{2} - \cos{\frac{\pi}{3}}\). Also, \(\cos{\frac{\pi}{3}} = \frac{1}{2}\). The function simplifies to \(f(\frac{\pi}{6}) = \sqrt{3} - \frac{1}{2}\).

Step 3: Convert to a single fraction

Finally, convert the expression to a single fraction by finding a common denominator. The common denominator of \(\sqrt{3}\) and \(\frac{1}{2}\) is 2. So, convert \(\sqrt{3}\) to \(\frac{2\sqrt{3}}{2}\). This gives: \(f(\frac{\pi}{6}) = \frac{2\sqrt{3} - 1}{2}\).

Step 4: Final Answer

So the exact value of the function with \(\theta = \frac{\pi}{6}\) is \(f(\frac{\pi}{6}) = \frac{2\sqrt{3} - 1}{2}\).

Key Concepts

Understanding the Cosine Function

The cosine function is a fundamental trigonometric function representing the ratio of the adjacent side to the hypotenuse in a right triangle. This means, for any angle \( \theta \), the cosine value expresses how much of the angle projects along the x-axis if you imagine the angle's terminal side starting from the origin. The cosine function is periodic with a period of \( 2\pi \), repeating its values every \( 360^{\circ} \). These values range from -1 to 1.The cosine values for some key angles such as \( \frac{\pi}{6} \), \( \frac{\pi}{3} \), and others can be critical for evaluating trigonometric expressions without a calculator. For example, \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Recognizing these common angles helps simplify many problems quickly. Memorizing these key cosine values helps you solve trigonometric problems effectively.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting known values of angles and simplifying the expression step by step. In our problem, the function given is \( f(\theta)=2 \cos \theta-\cos 2 \theta \). By substituting \( \theta = \frac{\pi}{6} \), we specifically calculate \( \cos \frac{\pi}{6} \) as \( \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} \) as \( \frac{1}{2} \). After substituting, simplify the expression:

• First multiply by any numerical coefficients.
• Combine your trigonometric terms correctly.

This method reduces errors and leads towards an accurate answer. It's crucial to follow precise steps and verify calculations during substitution and simplification stages.

Fraction Conversion and Simplification

Fraction conversion is the process of transforming an expression into a single fraction with a common denominator, allowing for cleaner representation and easier comparisons. Once the trigonometric value is evaluated, one common task is to convert the expression into a single fraction. In this exercise, after evaluating the expression \[ \sqrt{3} - \frac{1}{2} \], converting it to a unified format demands finding a common denominator, which in this case is 2.

• Represent \( \sqrt{3} \) as \( \frac{2\sqrt{3}}{2} \).
• Combine \( \frac{2\sqrt{3}}{2} - \frac{1}{2} \) into one term: \( \frac{2\sqrt{3} - 1}{2} \).

Each term must match the chosen common denominator before combining into a single fraction. This is a widely used technique in simplifying mathematical expressions for clearer and more concise results.