Problem 378 - Precalculus

Question

If \(\csc t=0.68,\) what is the \(\csc (-t) ?\)

Step-by-Step Solution

Answer

\( \csc(-t) = -0.68 \).

Step 1: Understanding Cosecant Function

The cosecant function, denoted as \( \csc t \), is defined as the reciprocal of the sine function. Thus, \( \csc t = \frac{1}{\sin t} \). Consequently, if \( \csc t = 0.68 \), we can infer that \( \sin t = \frac{1}{0.68} \).

Step 2: Identifying Trigonometric Identities

We need to determine how \( \csc(-t) \) is related to \( \csc t \). Using the identity for sine, \( \sin(-t) = -\sin t \), we can deduce that \( \csc(-t) = \frac{1}{\sin(-t)} = \frac{1}{-\sin t} \).

Step 3: Finding \( \csc(-t) \)

Given that \( \csc t = 0.68 \), we know \( \sin t = \frac{1}{0.68} \). Therefore, \( \sin(-t) = -\sin t = -\frac{1}{0.68} \). So, \( \csc(-t) = \frac{1}{-\sin t} = -0.68 \).

Key Concepts

Trigonometric Identities

Trigonometric identities form the foundation of understanding the relationships between trigonometric functions. These identities are equations involving trigonometric functions that are true for any value of the involved variables. They simplify expressions and solve complex problems by revealing fundamental relationships. For example, one key identity is the **even-odd identities** which helps in finding functions of negative angles.
- An even function, like cosine, satisfies the equation \( \cos(-x) = \cos(x) \).
- Conversely, an odd function, such as the sine function, satisfies \( \sin(-x) = -\sin(x) \).
This information is crucial when working with functions like the **cosecant**, which is the reciprocal of sine.
When given \( \csc t = 0.68 \), using trigonometric identities, we find \( \csc(-t) = -0.68 \) because \( \sin(-t) = -\sin(t) \). Recognizing and applying these identities simplifies the process of solving such problems.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions derive from the basic trigonometric functions by taking their reciprocals. These are:

• Cosecant (\( \csc \)), the reciprocal of sine (\( \sin \))
• Secant (\( \sec \)), the reciprocal of cosine (\( \cos \))
• Cotangent (\( \cot \)), the reciprocal of tangent (\( \tan \))

Understanding these reciprocal relationships is essential for converting between different trigonometric forms. The \( \csc t = \frac{1}{\sin t} \) defines the cosecant function, indicating it's undefined where the sine is zero.
The problem at hand involved finding \( \csc(-t) \).
By knowing \( \csc t = 0.68 \), the sine of \( t \) is calculated as \( \sin t = \frac{1}{0.68} \). Since \( \csc(-t) = \frac{1}{-\sin t} \), we find it to be \( -0.68 \). Such manipulations demonstrate the practical utility of understanding reciprocals in trigonometry.

Sine Function

The sine function is a fundamental trigonometric function that arises often in wave patterns and oscillatory motion. Defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle, these ratios can be understood on the unit circle as well.
When observing the sine function:

• The range is between -1 and 1.
• The sine of an angle \( t \), \( \sin t \), fulfills the equation for odd functions: \( \sin(-t) = -\sin(t) \).

This characteristic of sine makes it uniquely predictable and useful. For example, in the problem, understanding that \( \sin(-t) = -\sin t \) directly needed to be applied. This helped in determining \( \csc(-t) \) given a known \( \csc t = 0.68 \). Recognizing that \( \sin t \) equals \( \frac{1}{0.68} \), then \( \sin(-t) \) corresponds to its negative. Thus, appreciating the sine function's role and properties aids significantly in tackling trigonometric problems effectively.