Problem 11 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \cot 90^{\circ} $$

Step-by-Step Solution

Answer

The cotangent of 90 degrees is undefined.

Step 1: Understand the function

Cotangent is the reciprocal of tangent. Written as a formula, you can express this as \(\cot(x) = \frac{1}{\tan(x)}\). You'll use this formula to find the cotangent of 90 degrees.

Step 2: Find the tangent of the angle

This step involves knowing the values of common angles offhand. The tangent of 90 degrees is undefined, as tan(90 degrees) = sin(90 degrees)/cos(90 degrees), and the cosine of 90 degrees is 0, which would make you divide by zero.

Step 3: Find the cotangent of the angle

Since the cotangent is the reciprocal of the tangent, and the tangent of 90 degrees is undefined, the cotangent of 90 degrees is also undefined, because you can't take the reciprocal of an undefined value.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in connecting angles with ratios of sides in right triangles. They form the basis of understanding circles and waves in math and science. The primary trigonometric functions include:

• Sine (\( ext{sin}\))
• Cosine (\( ext{cos}\))
• Tangent (\( ext{tan}\))
• Cosecant (\( ext{csc}\))
• Secant (\( ext{sec}\))
• Cotangent (\( ext{cot}\))

Each function relates to an angle in a right-angled triangle and a specific ratio of its sides. For example, the sine of an angle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse.
These functions are critical not just in geometry but also in analyzing oscillations, waves, and circular motion.

Exploring Reciprocal Identities

Reciprocal identities are expressions that relate pairs of trigonometric functions. They essentially tell us how to find a function using another. Here’s how each pair works:

• \( ext{csc}(x) = \frac{1}{ ext{sin}(x)}\)
• \( ext{sec}(x) = \frac{1}{ ext{cos}(x)}\)
• \( ext{cot}(x) = \frac{1}{ ext{tan}(x)}\)

Understanding reciprocity helps in solving equations where values of one function lead to values of another. In the exercise, the cotangent is the reciprocal of the tangent. This means knowing the tangent allows us to determine the cotangent—if the tangent is defined. However, if the tangent is undefined, its reciprocal will also be undefined. This key concept helps simplify complex problems, especially when dealing with angles like 90 degrees.

Grasping Undefined Expressions

An expression becomes undefined when it involves a division by zero. In math, division by zero does not have a meaningful result. Consider the exercise with \( ext{tan}(90^{\circ})\).

• The tangent function is \( ext{tan}(x) = \frac{ ext{sin}(x)}{ ext{cos}(x)}\).
• At 90 degrees, \( ext{sin}(90^{\circ}) = 1\) and \( ext{cos}(90^{\circ}) = 0\).
• This leads to division by zero, making \( ext{tan}(90^{\circ})\) undefined.

As a result, since \( ext{cot}(x) = \frac{1}{ ext{tan}(x)}\), the cotangent at 90 degrees is also undefined.
Recognizing undefined expressions is crucial in avoiding mathematical errors and is essential for solid problem-solving skills.