Problem 10 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \cot 60^{\circ} $$

Step-by-Step Solution

Answer

\(\cot 60^{\circ} = \frac{\sqrt{3}}{3}\)

Step 1: Understand the cotangent

The cotangent of an angle is defined as the reciprocal of the tangent of that angle. In mathematical terms, if \( y = \tan(x) \), then \( \cot(x) = \frac{1}{y} = \frac{1}{\tan(x)} \).

Step 2: Find the tangent of 60 degrees

The tangent of 60 degrees is \(\sqrt{3}\). This can be found using a special right triangle or the unit circle.

Step 3: Find the cotangent of 60 degrees

Since the cotangent is the reciprocal of the tangent, and the tangent of 60 degrees is \(\sqrt{3}\), the cotangent of 60 degrees is \( \frac{1}{\sqrt{3}} \). But, we usually rationalize the denominator, to make the final answer \(\frac{\sqrt{3}}{3}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the ratios of its sides. They are fundamental in studying geometry, oscillations, and other phenomena. In trigonometry, we mainly work with six functions, namely sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each function serves as a crucial building block for understanding the properties of angles and their relationships.

• **Sine** deals with the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• **Cosine** involves the ratio of the adjacent side to the hypotenuse.
• **Tangent** focuses on the ratio of the opposite side to the adjacent side, given by \(\frac{\sin}{\cos}\).
• **Cotangent** is the reciprocal of tangent, defined as \(\frac{1}{\tan}\) or \(\frac{\cos}{\sin}\).

Trigonometric functions permit the calculation and prediction of unknown lengths and angles. They’re essential in fields like physics, engineering, and even computer graphics.

Special Angles

Special angles are specific angles commonly used in trigonometry due to their neat properties and simple, rational trigonometric values. Mastery over these angles is advantageous as they frequently appear in trigonometry problems and serve as benchmarks for other calculations. Important special angles are 30°, 45°, 60°, and their multiples.

• **30°**, **45°**, and **60°** are not just arbitrary angles. They correspond to angles found in equilateral and isosceles right triangles, which exhibit innate symmetry.

When analyzing these angles:

• The tangent of 60°, for instance, is \(\sqrt{3}\) because it arises from a 30°-60°-90° triangle where the sides are in a ratio of 1: \(\sqrt{3}\): 2.
• Knowing these ratios helps us find values quickly. For 60°, while tangent gives us \(\sqrt{3}\), cotangent provides the reciprocal value, \(\frac{1}{\sqrt{3}}\).

Understanding these relationships simplifies work with angles in trigonometric expressions.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical process used to simplify fractions. When a denominator contains an irrational number, often a square root, it can make calculations cumbersome, so we rationalize it. This involves removing the square root from the denominator by multiplying both the numerator and the denominator by a conjugate if necessary.

• For example, when we find that \(\cot(60^{\circ}) = \frac{1}{\sqrt{3}}\), we prefer to express it in a rational form.
• We multiply both the numerator and the denominator by \(\sqrt{3}\) to get \(\frac{\sqrt{3}}{3}\)

This makes the expression easier to understand and manipulate. Rationalizing the denominator is critical when dealing with trigonometric expressions, ensuring clarity and consistency during further calculations.