Problem 89 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each expression without using a calculator. $$\tan \frac{\pi}{3}-\cos \frac{\pi}{6}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(\sqrt{3}/2\).

Step 1: Identify the Trigonometric Values for Specific Angles

We need to remember the trigonometric values of \(\pi/3\) and \(\pi/6\). For \(\pi/3\) radians (or 60 degrees), the tangent value, \(\tan(\pi/3)=\sqrt{3}\). For \(\pi/6\) radians (or 30 degrees), the cosine value, \(\cos(\pi/6)=\sqrt{3}/2\).

Step 2: Substitute the Trigonometric Values in the Given Expression

Replace \(\tan(\pi/3)\) with \(\sqrt{3}\), and replace \(\cos(\pi/6)\) with \(\sqrt{3}/2\) in the given expression. The modified expression becomes: \(\sqrt{3} - \sqrt{3}/2\).

Step 3: Simplify the Expression

To subtract these terms, write them with a common denominator, which in this case would be 2. So, \(\sqrt{3}\) becomes \((2\sqrt{3})/2\) and our equation is now \((2\sqrt{3})/2 - (\sqrt{3}/2)\), which simplifies to \((\sqrt{3}/2)\).

Key Concepts

Tangent of Pi/3

When dealing with trigonometry, it's essential to know the value of the tangent function for commonly used angles, such as \(\frac{\text{π}}{3}\) (or 60 degrees). The \(\tan\left(\frac{\text{π}}{3}\right)\) equals \(\sqrt{3}\). This is derived from the equilateral triangle where each angle is 60 degrees.

To visualize this without a calculator, imagine splitting an equilateral triangle into two right triangles by drawing a line from one vertex to the midpoint of the opposite side. The sides of the resulting right triangle are now in a 1:2:\(\text{sqrt}{3}\) ratio. The tangent of an angle is the ratio of the length of the opposite side to the adjacent side. In this case, for the 60-degree angle, it is \(\sqrt{3}\).

Understanding this fundamental value is crucial for solving trigonometry problems involving the tangent function.

Cosine of Pi/6

In a similar fashion, recognizing the cosine of \(\frac{\text{π}}{6}\) or 30 degrees is equally important. The value is \(\frac{\text{sqrt}{3}}{2}\). This is one of the standard trigonometric values often memorized through special triangles or the unit circle.

For a 30-degree angle in a 30-60-90 triangle, the longest side (the hypotenuse) is twice the length of the shortest side (opposite the 30-degree angle), with the remaining side (adjacent to the 30-degree angle) being \(\text{sqrt}{3}\) times the shortest side. The cosine function represents the length of the adjacent side divided by the hypotenuse, which in this case will be the ratio of \(\text{sqrt}{3}\) to 2.

Trigonometry Without Calculator

Performing trigonometry without the use of a calculator can seem daunting, but it can be managed quite effectively by understanding key concepts, memorizing certain values, and learning to visualize trigonometric ratios.

• Memorize the trigonometric ratios for common angles such as 0°, 30°, 45°, 60°, and 90°.
• Understand the unit circle and how the coordinates correspond to cosine and sine values.
• Remember the special right triangles, which can provide trigonometric ratios without calculations.
• Practice simplifying trigonometric expressions, as this is often necessary to find exact values.

These techniques are fundamental in developing the ability to calculate trigonometric values in your head and not relying entirely on technology.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a skill that can make solving trigonometry problems more manageable. The process involves a few steps:

• Identify and apply fundamental trigonometric identities.
• Break down complex expressions into simple ones by using algebraic manipulations.
• Factor where possible to combine like terms or simplify expressions.
• When subtracting fractions, as in the exercise above, find a common denominator to combine the terms effectively.

By simplifying expressions, not only are the results more elegant, but the necessary steps to solve a problem are often revealed. Grasping these concepts helps to enhance problem-solving skills in trigonometry.