Problem 119 Use a calculator to demonstrate ... [FREE SOLUTION]

Chapter 5: Problem 119

Use a calculator to demonstrate the identity for each value of \(\boldsymbol{\theta}\). \(\csc ^{2} \theta-\cot ^{2} \theta=1\) (a) \(\theta=132^{\circ}\) (b) \(\theta=\frac{2 \pi}{7}\)

Short Answer

Expert verified

For both \(\theta = \frac{22\pi}{30}\) and \(\theta = \frac{2\pi}{7}\), the calculated result of \( \csc^2 \theta - \cot^2 \theta \) is 1, thus verifying the identity for these angles.

Step by step solution

Convert degrees to radians

First, convert the angle from degrees to radians if necessary. In (a) where \(\theta = 132掳\), we use the conversion factor \( \frac{\pi}{180}\) to convert degrees to radians: \(\theta = 132掳 * \frac{\pi}{180} = \frac{22\pi}{30} \). In (b), the angle \(\theta = \frac{2\pi}{7}\) is already given in radians.

Calculate cosecant and cotangent

Next, calculate the cosecant and cotangent of the angles. The cosecant is \(\csc \theta = 1/ \sin \theta\) and the cotangent is \(\cot \theta = \cos \theta / \sin \theta\). Use a calculator to find the sine and cosine of the angles, and then calculate the cosecant and cotangent.

Verify the identity

Now, square the values of cosecant and cotangent, subtract cotangent squared from cosecant squared and verify that the result is 1. For each angle, if \( \csc^2 \theta - \cot^2 \theta = 1 \), then the identity is verified for that angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Cosecant Function

The cosecant function, often written as \( \csc \theta \), is the reciprocal of the sine function. It means that cosecant is defined as \( \csc \theta = \frac{1}{\sin \theta} \). This function is important in trigonometry because it offers another way to express relationships between the sides of a right triangle. Here's what you need to know:

When \( \sin \theta = 0 \), the cosecant is undefined, because division by zero is not possible.
The graph of the cosecant function is characterized by a series of curves and vertical asymptotes where the sine function is zero.
Cosecant values are always greater than or equal to 1 or less than or equal to -1. This is because sine is always between -1 and 1.<\/ul>Understanding the properties of the cosecant function can help in solving complex trigonometric identities, such as \( \csc^2 \theta - \cot^2 \theta = 1 \). Knowing the basic definitions and properties is crucial for verifying identities or evaluating expressions that involve this function.

Diving into the Cotangent Function

The cotangent function is another important reciprocal trigonometric function. It is defined as the reciprocal of the tangent function: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Here are some key aspects to remember about cotangent:

The cotangent is undefined where \( \sin \theta = 0 \), as division by zero is impossible.
The graph of \( \cot \theta \) alternates between positive and negative values and includes vertical asymptotes wherever the sine function is zero.
Cotangent values depend heavily on the fact that it combines both cosine and sine. The behavior of both functions affects the cotangent.
Cotangent is useful in various trigonometric identities and equations. An example is verifying identities like \( \csc^2 \theta - \cot^2 \theta = 1 \), demonstrating relationships between trigonometric functions.<\/ul>Having a clear grasp of how cotangent operates alongside other trigonometric functions can greatly assist in solving trigonometric identities and equations effectively.

The Degree-to-Radian Conversion

Trigonometry often requires converting angles from degrees to radians because radians are a more natural measure in mathematics. One degree equals \( \frac{\pi}{180} \) radians. Here's a quick guide:

To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \). This allows you to work with trigonometric functions in their most commonly used units.
Radians provide a direct link to arc length and the unit circle, simplifying many mathematical calculations.
For instance, converting \(132^\circ\) gives \(132 \times \frac{\pi}{180}\), which simplifies to \(\frac{22\pi}{30}\) radians.
This conversion is necessary for many calculations and functions, especially those involving identity verifications such as \( \csc^2 \theta - \cot^2 \theta = 1 \).

Understanding this conversion process will enhance your ability to solve trigonometric problems accurately in both theoretical and practical contexts.

91影视

Use a calculator to demonstrate the identity for each value of \(\boldsymbol{\theta}\). \(\csc ^{2} \theta-\cot ^{2} \theta=1\) (a) \(\theta=132^{\circ}\) (b) \(\theta=\frac{2 \pi}{7}\)

Short Answer

Step by step solution

Convert degrees to radians

Calculate cosecant and cotangent

Verify the identity

Key Concepts

Understanding the Cosecant Function

Diving into the Cotangent Function

The Degree-to-Radian Conversion

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