Problem 14 - Algebra and Trigonometry

Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator. $$ \cot \frac{\pi}{3} $$

Step-by-Step Solution

Answer

The value of \( \cot \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{3} \)

Step 1: Understanding the term

Firstly, it's important to remember that the cotangent of an angle in a right triangle is defined as the ratio of the side adjacent to the angle to the side opposite the angle. In this case, the angle is \( \frac{\pi}{3} \) radians, which is equivalent to 60 degrees in the unit circle.

Step 2: Evaluate the Cotangent

For a 60-degree angle in the unit circle, the adjacent side is 1/2 and the opposite side is \(\sqrt{3}/2\). Therefore, cotangent,\( \cot \frac{\pi}{3} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \)

Step 3: Rationalizing the denominator

To express the value without a square root in the denominator, we need to rationalize the denominator. This is accomplished by multiplying the numerator and denominator by \(\sqrt{3}\). Thus, \( \frac{1}{\sqrt{3}} = \frac{1*\sqrt{3}}{\sqrt{3}*\sqrt{3}} = \frac{\sqrt{3}}{3} \)

Key Concepts

Cotangent

The cotangent is one of the six fundamental trigonometric ratios. It's often symbolized as "cot." In a right triangle, the cotangent of an angle is calculated as the ratio of the adjacent side to the opposite side.
For the angle \( \frac{\pi}{3} \) , which corresponds to 60 degrees in a circle, the cotangent can be found using these sides:

• The side adjacent to the angle \( \frac{\pi}{3} \) is typically \( \frac{1}{2} \).
• The side opposite the angle is \( \frac{\sqrt{3}}{2} \).

The formula for cotangent tells us to divide the adjacent side by the opposite side. To solve for \( \cot \frac{\pi}{3} \) , you would use: \[ \cot \frac{\pi}{3} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \] This shows how cotangent is directly connected to these trigonometric side ratios.

Unit Circle

The unit circle is a vital tool in understanding and visualizing trigonometric functions and their values at various angles. This circle is called the "unit" circle because its radius is exactly one unit.
Key aspects of the unit circle include:

• It is centered at the origin \( (0, 0) \).
• The radius is \( 1 \), making trigonometric calculations straightforward because the hypotenuse in reference triangles is always \( 1 \).

Using the unit circle, we can derive the angles in radians, such as \( \frac{\pi}{3} \), and relate them to the \[(x, y)\] coordinates. Each point on the circle corresponds to an angle measured from the positive x-axis, helping to visually represent \(\sin\), \(\cos\), and \(\tan\) functions, amongst others. For \( \frac{\pi}{3} \), 60 degrees, we generally find the \(x\) -coordinate \( \frac{1}{2} \) , and the \(y\) -coordinate \( \frac{\sqrt{3}}{2} \). This information allows us to conveniently compute other trigonometric ratios like cotangent.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots or irrational numbers in the denominator of a fraction. This process makes expressions easier to work with and compare, particularly for further mathematical operations.
Here’s a step-by-step guide to rationalizing:

• Start with the fraction with a square root in the denominator, say \( \frac{1}{\sqrt{3}} \).
• Multiply both the numerator and the denominator by the same square root, here \( \sqrt{3} \), which cancels out in the denominator.
• The operation \((\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}})\) transforms the fraction to \( \frac{\sqrt{3}}{3} \).

Being proficient in this technique is essential, as it helps achieve simpler, rational denominators in trigonometric calculations and various algebraic expressions, allowing easy addition or subtraction of such values.