Problem 111 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\sin \theta=\frac{1}{2}, \csc \theta=2$$

Step-by-Step Solution

Answer

The statement is possible since the values match.

Step 1: Understand the Definitions

First, remember the definitions of the trigonometric functions \( \sin \theta \) and \( \csc \theta \). The sine function \( \sin \theta \) gives the ratio of the opposite side to the hypotenuse in a right triangle. The cosecant function \( \csc \theta \) is the reciprocal of \( \sin \theta \), i.e., \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Calculate \( \csc \theta \) from \( \sin \theta \)

Given that \( \sin \theta = \frac{1}{2} \), we can find \( \csc \theta \) by taking the reciprocal: \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \).

Step 3: Compare \( \csc \theta \) Values

Both the given \( \csc \theta \) value and the calculated \( \csc \theta \) value are 2, which means they match exactly. Thus, the statement is possible.

Key Concepts

Sine function

The sine function is one of the six fundamental trigonometric functions. When we talk about the sine of an angle \( \theta \), we are typically referring to the ratio that exists in a right triangle. Specifically, the sine function is defined as the ratio of the length of the opposite side to the hypotenuse.
This means if you have a right-angled triangle and angle \( \theta \) is one of the non-right angles, then:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

The sine function is fundamental in studying periodic phenomena, such as waves. It is also crucial in various applications ranging from engineering to physics. When we say \( \sin \theta = \frac{1}{2} \), it tells us about a specific angle, which has specific values, such as 30° or \( \frac{\pi}{6} \) radians in a unit circle.
Understanding the sine function helps in getting a good grip on the behavior of angles and their respective ratios in any given triangle.

Cosecant function

The cosecant function, denoted \( \csc \theta \), is less commonly discussed than its counterpart, the sine function, but it plays an equally important role in trigonometry. The cosecant of an angle \( \theta \) is defined as the reciprocal of the sine function.

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

Since the sine function relates the opposite side to the hypotenuse in a triangle, the cosecant represents the ratio of the hypotenuse to the opposite side. This means for an angle \( \theta \) where \( \sin \theta = \frac{1}{2} \), the corresponding cosecant value would be:

• \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \)

This reveals the inherent link between the two functions, as they are directly proportional, provided that the sine of the angle is non-zero.

Reciprocal functions

Reciprocal functions are instrumental in trigonometry for transforming functions into their complementary forms. When you talk about reciprocal functions with respect to trigonometric functions, you are essentially flipping them.
For the sine function \( \sin \theta \), its reciprocal function is the cosecant \( \csc \theta \). They possess a reciprocal relationship:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• Similarly, the reciprocal of cosine is secant and for tangent, it is cotangent.

Reciprocal functions expand the range of trigonometric analysis and provide alternate methods to verify equations and solving triangles in trigonometry. Understanding reciprocal functions like cosecant, helps not only verify calculations but also explore properties of angles and triangles from a different perspective.
They become particularly useful when dealing with expressions and equations across different branches of mathematics and applied sciences, ensuring balanced and complete mathematical descriptions.