Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the value of the trigonometric function to evaluate each function. \(\sin t=\frac{1}{3}\) (a) \(\sin (-t)\) (b) \(\csc (-t)\)

Step-by-Step Solution

Answer

The result of the exercise is: (a) \(\sin (-t) = - \frac{1}{3}\), (b) \(\csc (-t) = 3\)

Step 1: Evaluating \(\sin (-t)\)

Use the property of sine that \(\sin (-t) = -\sin t\). This property tells us that the sine function is odd, meaning it has rotational symmetry about the origin. Therefore, \(\sin (-t) = -\sin t = - \frac{1}{3}\)

Step 2: Evaluating \(\csc(-t)\)

The cosecant of any angle is the reciprocal of the sine of that angle. Therefore, to find \(\csc (-t)\), we need to take the reciprocal of \(\sin (-t)\), i.e., \(\csc (-t) = -\csc t = -\frac{1}{\sin t} = -\frac{1}{-\frac{1}{3}} = 3\)

Key Concepts

Understanding the Sine Function

The sine function is a fundamental trigonometric function often used in various mathematical contexts. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This ratio varies as the angle changes, making it essential for describing oscillatory or wave-like phenomena.

The sine of an angle, typically expressed as \(\sin \theta\), is periodic with a period of \(2\pi\). This means its values repeat every \(2\pi\) radians. Understanding this periodic nature can help in solving problems related to cycles and rotations.

Another key aspect of the sine function is its range and domain. For \(-1 \leq \sin t \leq 1\), it is defined for all real numbers, making it versatile for many applications. When you see \(\sin t = \frac{1}{3}\), it implies the angle is such that sine equals this fractional value. Knowing how to compute and interpret these values is crucial for solving trigonometric problems.

Exploring Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are layers of trigonometry that provide deep insights when analyzing angles and triangles. These functions are simply the reciprocal values of the primary trigonometric functions.

• **Cosecant (\(\csc\))**: The reciprocal of the sine function, \(\csc \theta = \frac{1}{\sin \theta}\). If \(\sin t = \frac{1}{3}\), then \(\csc t = 3\).


• **Secant (\(\sec\))**: The reciprocal of the cosine function.


• **Cotangent (\(\cot\))**: The reciprocal of the tangent function.

Understanding these reciprocals helps with more complex trigonometric calculations. When given values like \(\sin t = \frac{1}{3}\), finding \(\csc t\) involves simply taking the reciprocal, making it useful for dealing with angles that aren’t standard. This understanding is particularly helpful in solving problems where memorizing specific angle functions might not be feasible.

Discovering Odd Functions

Odd functions exhibit unique symmetry characteristics. A function \(f(x)\) is considered odd if \(f(-x) = -f(x)\) for all \(x\). This means the graph of the function has rotational symmetry around the origin.

The sine function \(\sin(t)\) is a classic example of an odd function. This property is what allows us to state \(\sin(-t) = -\sin(t)\).

This understanding of odd functions can simplify computations in mathematical problems. When you come across equations involving \(\sin(-t)\), knowing it equals \(-\sin(t)\) helps solve such problems with ease.

Being equipped with the knowledge of odd and even functions broadens one’s ability to analyze and interpret various trigonometric expressions effectively. Thus, understanding odd functions is pivotal for tackling diverse mathematical scenarios efficiently.