Problem 56 - Algebra 2

Question

Writing Explain why each expression is undefined. $$ \csc 180^{\circ} $$

Step-by-Step Solution

Answer

The expression \(\csc(180^{\circ})\) is undefined because it results in division by zero, which is undefined in mathematics.

Step 1: Understand the cosecant function

The cosecant function, often written as \(\csc(\theta)\), is defined as the reciprocal of the sine function: \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Now, the sine function ranges between -1 and 1, and certain angles give a sine value of 0 (example: 0°, 180°, 360°, and so on).

Step 2: Analyze the given angle

In this case, the angle given is 180°. The sine of 180° is 0. This comes from the unit circle representation, where at 180°, the y-coordinate (which is representative of the sine value) is 0.

Step 3: Reason why the expression is undefined

Substituting the value of \(\sin(180^{\circ})\) into the equation of \(\csc(\theta) = \frac{1}{\sin(\theta)}\) gives \(\csc(180^{\circ}) = \frac{1}{0}\). Division by zero is undefined in mathematics, so \(\csc(180^{\circ})\) is undefined.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental part of trigonometry. It's represented as \( \sin(\theta) \), where \( \theta \) is an angle. Sine measures the vertical component or the y-coordinate of a point on the unit circle.

• It oscillates between -1 and 1 as \( \theta \) varies.
• On the unit circle, sine reaches its maximum value of 1 at 90° and a minimum value of -1 at 270°.
• When \( \theta = 180° \), the sine value is 0 because the point on the unit circle aligns with the x-axis.

Recognizing these key values and how they relate to different angles is crucial for solving many trigonometric problems.

Exploring Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every valid input. These identities help us simplify and manipulate trigonometric expressions.

• Reciprocal identities are highly useful. For instance, the cosecant function is the reciprocal of the sine function: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• Other fundamental identities include \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

These identities are essential tools for solving trigonometric problems and functions, especially when dealing with undefined expressions or complex calculations.

Understanding the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a crucial concept in trigonometry used for defining the trigonometric functions.

• Each point on the unit circle can be represented as \( (\cos(\theta), \sin(\theta)) \).
• At 0° and 360°, the coordinate is (1, 0), where \( \sin \) is 0.
• At 90°, the coordinate is (0, 1), where \( \sin \) is 1.
• At 180°, the coordinate is (-1, 0), where \( \sin \) is 0 once again.

This concept helps visualize how angles correspond to specific sine and cosine values, which makes it easier to comprehend expressions and their evaluations.

Explaining Undefined Expressions

An expression becomes undefined when its mathematical evaluation leads to a scenario that doesn't make sense. This often occurs in division by zero situations.

• When calculating \( \csc(180°) \), we find that \( \sin(180°) = 0 \).
• Substituting in the reciprocal identity, \( \csc(180°) = \frac{1}{0} \), results in an undefined expression.
• Division by zero is undefined because there's no number that can multiply with zero to produce one.

Understanding when and why expressions are undefined is crucial in mathematics, as it helps prevent errors in calculations and interpretations.