Problem 86 - Precalculus

Question

Use a calculator to approximate cse \(211^{\circ} .\) What do you expect \(\csc \left(-211^{\circ}\right)\) to be? Verify your answer with a calculator.

Step-by-Step Solution

Answer

Approximate values: \(\csc 211^{\circ} \approx -2.13\), \(\csc(-211^{\circ}) \approx 2.13\).

Step 1: Review Definitions

The cosecant function, denoted as \(\csc \theta\), is the reciprocal of the sine function: \(\csc \theta = \frac{1}{\sin \theta}\). We are going to compute \(\csc 211^{\circ}\) using this definition.

Step 2: Calculate \(\sin 211^{\circ}\)

Use your calculator to find \(\sin 211^{\circ}\). Ensure your calculator is in degree mode. The sine of \(211^{\circ}\) is approximately \(-0.4695\).

Step 3: Compute \(\csc 211^{\circ}\)

Using the reciprocal relationship, calculate \(\csc 211^{\circ} = \frac{1}{\sin 211^{\circ}} = \frac{1}{-0.4695}\). This gives us \(\csc 211^{\circ} \approx -2.13\).

Step 4: Understand the Significance of Negative Angles

The cosecant function, like the sine function, is odd. This means \(\csc(-\theta) = -\csc(\theta)\). Therefore, \(\csc(-211^{\circ}) = -\csc(211^{\circ})\).

Step 5: Compute \(\csc(-211^{\circ})\) and Verify

From the earlier calculation, \(\csc 211^{\circ} \approx -2.13\), so \(\csc(-211^{\circ}) = -(-2.13) \approx 2.13\). Verify by calculating \(\sin(-211^{\circ})\) which gives approximately \(-0.4695\), then \(\csc(-211^{\circ}) = \frac{1}{-(-0.4695)} = 2.13\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical relationships used to study relationships involving angles and distances in triangles. These functions are essential in many fields of science and engineering, providing key insights into phenomena that involve periodic occurrences. For any angle, the six main trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Each function has a specific relationship with a right triangle’s sides.
The cosecant function is particularly interesting as it is less commonly used compared to others like sine or cosine. It is represented as \(\csc \theta\), where \(\theta\) is the angle of interest. The cosecant of an angle, \(\csc \theta\), is defined as the ratio of the hypotenuse to the opposite side in a right-angled triangle, or simply the reciprocal of the sine of that angle: \[\csc \theta = \frac{1}{\sin \theta}.\] Understanding these functions and their reciprocal relationships is crucial for solving trigonometric equations effectively.

Reciprocal Identity

The reciprocal identity is one of the fundamental identities in trigonometry, assisting in the understanding of different trigonometric functions. It states that certain trigonometric functions are the reciprocals of others. For instance, the sine and cosecant functions are reciprocals.
This reciprocal relationship can be expressed mathematically as \[\csc \theta = \frac{1}{\sin \theta}.\] This means if you know the value of sine for an angle, the cosecant is simply the inverse of that value.

• If \(\sin 211^{\circ} = -0.4695\), then \(\csc 211^{\circ} = \frac{1}{-0.4695}\), which is approximately \(-2.13\).
• This relationship helps us quickly determine values for less common functions by using more familiar ones.

Using reciprocal identities is a powerful tool to navigate through different trigonometric problems, simplifying calculations and understanding of angle measurements.

Negative Angles

When dealing with trigonometric functions, understanding the behavior of negative angles is crucial. Negative angles are often encountered in mathematical and physical problems where directions are reversed or when measuring clockwise from a point.
It's important to know that many trigonometric functions exhibit specific behaviors with negative angles. For example, sine and cosecant functions are odd functions, which means they show symmetric properties around the origin on a graph. Mathematically, this is expressed as \[\sin(-\theta) = -\sin(\theta)\] and \[\csc(-\theta) = -\csc(\theta).\]

• For the angle \(211^{\circ}\), if \(\csc 211^{\circ} \approx -2.13\), then \(\csc(-211^{\circ}) = -(-2.13) = 2.13\).
• This means the value of the cosecant changes sign when changing the angle's sign.

Understanding how negative angles affect trigonometric functions ensures precision in calculations and insights into how these functions behave, which is paramount for accurate problem-solving.