Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \csc 180^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \(\csc 180^{\circ}\) is undefined.

Step 1: Understanding the Function

The cosecant function, denoted as \(\csc\), is the reciprocal of the sine function. Therefore, \(\csc \theta = \frac{1}{\sin \theta}\). So we need to find \(\sin 180^{\circ}\) first.

Step 2: Evaluate \(\sin 180^{\circ}\)

The sine of 180 degrees is given by \(\sin 180^{\circ} = 0\). This is because at 180 degrees, the sine function value on the unit circle is at the origin horizontally with no vertical component.

Step 3: Apply the Cosecant Formula

Now substitute \(\sin 180^{\circ} = 0\) into the cosecant formula: \(\csc 180^{\circ} = \frac{1}{\sin 180^{\circ}} = \frac{1}{0}\).

Step 4: Determine the Result

Since division by zero is undefined in mathematics, the value of \(\csc 180^{\circ}\) is undefined.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in the study of geometry and wave applications. They are functions of an angle and are crucial in understanding the relationships in triangles.

• The most common trigonometric functions are sine, cosine, and tangent.
• These functions can help relate the angles and lengths of triangles.
• Trigonometric functions are periodic, meaning they repeat their values in regular intervals.

Each trigonometric function has a unique shape in a graph, often forming a wave-like pattern, which is why they are essential in modeling cyclical phenomena such as sound waves.
Understanding these functions provides crucial insights into the behavior of angles and allows us to solve problems involving triangles effectively.

Exploring the Sine Function

The sine function is one of the primary trigonometric functions. It relates a right triangle's angle to the ratio of its opposite side to its hypotenuse.

• The sine function is denoted by \(\sin\).
• It is periodic with a cycle of 360 degrees or \(2\pi\) radians.
• At specific angles like 0, 90, 180, and 270 degrees, the sine function takes distinct values.

At 180 degrees, the sine value is 0. This is because 180 degrees on the unit circle lands on the negative x-axis, where the vertical (y) component, which sine measures, is zero.
By understanding the sine function's values at key angles, you can solve a range of trigonometric problems effectively.

Connecting Through Reciprocal Identities

Reciprocal identities are used to express one trigonometric function in terms of another via reciprocal relationships. The cosecant function (\(\csc\)) is the reciprocal of the sine function.

• The formula is \(\csc \theta = \frac{1}{\sin \theta}\).
• This relationship shows how close trigonometric functions can work together to solve problems.
• When sine is zero, as in the case of \(\sin 180^{\circ}\), cosecant becomes undefined.

It's important to remember that division by zero is invalid in any context, so if ever the sine of an angle is zero, the cosecant will be undefined.
Recognizing the reciprocal identities allows for better problem-solving in trigonometry and can aid in simplifying complex trigonometric expressions.