Problem 18 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \csc 30^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \csc 30 \) is 2.

Step 1: Recalling Sine Function Value

First, recall that the sine of 30 degrees is \( \sin 30 = \frac{1}{2} \).

Step 2: Find the Reciprocal

Since cosecant is the reciprocal of sine, find the reciprocal value of \( \sin 30 \). That is, \( \csc 30 = \frac{1}{\sin 30} \).

Step 3: Substitute the Value

Substitute \( \sin 30 = \frac{1}{2} \) in the equation. Thus, \( \csc 30 = \frac{1}{\frac{1}{2}} \).

Key Concepts

Understanding Cosecant

The cosecant function is an essential trigonometric function often encountered in mathematics, particularly in trigonometry. Cosecant, abbreviated as "csc," is not as commonly used as sine or cosine, but it's important in various calculations. The cosecant of an angle in a right triangle is defined as the ratio of the length of the hypotenuse to the length of the opposite side.
In formula terms, for an angle \( \theta \), this is represented as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This definition implies that the cosecant is the reciprocal of the sine function. Understanding cosecant will allow you to easily navigate the relationship between these trigonometric functions. Remember, the cosecant values are undefined for angles where the sine is zero, such as 0 degrees and multiples of 180 degrees.

The Nature of Sine

The sine function is one of the foundational trigonometric functions. It arises in the study of right triangles and circles, particularly characterized by its wave-like nature. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

• For instance, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

Sine values are pivotal because they oscillate between -1 and 1, giving them wide applicability in physics and engineering, among other fields. For specific angles, sine has known values, such as \( \sin 30^{\circ} = \frac{1}{2} \). Knowing these values helps in solving various trigonometric problems effectively. Sine also plays a critical role in the unit circle, where it reflects the y-coordinate of a point on the circle corresponding to a given angle.

Exploring the Reciprocal Function

Reciprocal functions are functions that "flip" another function by expressing its inverse in terms of division. In trigonometry, reciprocity comes into play when dealing with functions like sine, cosine, and tangent.The reciprocal of a function \( f(x) \) is \( \frac{1}{f(x)} \). For sine, this gives us the cosecant:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Reciprocal functions are particularly useful when original function values are challenging to interpret or utilize directly. By inverting the function, you can gain new insights or simplify calculations.It is crucial to note, however, that when the function value approaches zero, the reciprocal can become undefined or grow infinitely large. This is why we need to identify cases, like the sine of certain angles, where the cosecant function becomes undefined. Keeping this in mind empowers you to handle trigonometric problems with greater finesse.