Problem 119 - Precalculus with Limits

Question

In Exercises 119 - 122, use a calculator to demonstrate the identity for each value of \( \theta \). \( \csc^2 \theta - \cot^2 \theta = 1 \) (a) \( \theta = 132^\circ \) (b) \( \theta = \dfrac{2\pi}{7} \)

Step-by-Step Solution

Answer

The identity \( \csc^2 \theta - \cot^2 \theta = 1 \) is shown to hold true for the given angles \( \theta = 132^\circ \) and \( \theta = \dfrac{2\pi}{7} \).

Step 1: Convert degree to radian (only for first value)

First, convert the degrees to radians since the calculator functions typically used in trigonometry are in radians. Use the formula \( rad = degrees \times \dfrac{\pi}{180} \). So, \( \theta = 132^\circ = 132 \times \dfrac{\pi}{180} = \dfrac{22\pi}{30} \).

Step 2: Calculate cosecant and cotangent for \( \theta = 132^\circ \)

Now, use the calculator to find the value of \( \csc^2 \theta \) and \( \cot^2 \theta \) with \( \theta \) being \( \dfrac{22\pi}{30} \). Remember \( \csc \theta = \dfrac{1}{\sin \theta} \) and \( \cot \theta = \dfrac{\cos \theta}{\sin \theta} \). Calculate these values and then subtract \( \cot^2 \theta \) from \( \csc^2 \theta \).

Step 3: Verify the identity for \( \theta = 132^\circ \)

The result after the subtraction should be as close as possible to 1, thus demonstrating the identity for this angle.

Step 4: Calculate cosecant and cotangent for \( \theta = \dfrac{2\pi}{7} \)

Now, use the calculator to find the value of \( \csc^2 \theta \) and \( \cot^2 \theta \) with \( \theta \) being \( \dfrac{2\pi}{7} \). Calculate these values and then subtract \( \cot^2 \theta \) from \( \csc^2 \theta \).

Step 5: Verify the identity for \( \theta = \dfrac{2\pi}{7} \)

Again, the result after the subtraction should be as close as possible to 1, thus demonstrating the identity for this angle.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the trigonometric functions derived from the sine function. It is defined as the reciprocal of the sine function. This means:

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

Since the sine function, \( \sin \theta \), represents the ratio of the opposite side to the hypotenuse in a right-angled triangle, the cosecant function represents the ratio of the hypotenuse to the opposite side.
By considering the values where sine equals zero, you can note that cosecant is undefined at these points, because division by zero is undefined. For instance, at \( \theta = 0, \pi, 2\pi, \ldots\), \( \csc \theta \) doesn't exist. Understanding the behavior of \( \csc \theta \) helps when verifying trigonometric identities and solving equations since it's closely involved with the unit circle properties. Pay attention that when \( \sin \theta \) approaches 1, \( \csc \theta \) also approaches 1, reinforcing the identity \( \csc^2 \theta - \cot^2 \theta = 1 \).

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is another reciprocal trigonometric function. It is defined relative to the tangent function, where:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

In a right triangle, tangent is the ratio of the opposite side to the adjacent side, so cotangent reverses it, representing the adjacent side to the opposite side.
This ratio signifies that \( \cot \theta \) is undefined where \( \tan \theta \) is zero, specifically at multiples of \( \pi/2 \), such as \( \pi/2, 3\pi/2, \ldots \). In these scenarios, \( \sin \theta = 0 \) leading \( \cot \theta \) to become undefined due to division by zero.
Understanding \( \cot \theta \) is crucial in trigonometric identity verifications and transformations. Recognizing the relationship between \( \cot \theta \) and other functions, as in \( \csc^2 \theta - \cot^2 \theta = 1 \), aids in identifying unit circle symmetry and periodicity in applications.

Radians and Degrees Conversion

Radians and degrees are two units of measuring angles in mathematics. Degrees are more common in day-to-day activities, whereas radians are typically used in mathematical and scientific contexts, particularly calculus and trigonometry.

• A full circle is \( 360^\circ \) in degrees, equivalent to \( 2\pi \) in radians.
• The conversion formula is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

This conversion is vital when using trigonometric functions in calculators, as these often default to radian mode. For example, converting \( 132^\circ \) to radians involves multiplying it by \( \frac{\pi}{180} \), resulting in \( \frac{22\pi}{30} \), which simplifies to \( \frac{11\pi}{15} \).
Understanding how to switch between these units is important not only for computation but also for comprehension when reading trigonometric graphs or identities. Consistency in the units used across calculations ensures correctness and coherence, especially in exercises demonstrating theorems like \( \csc^2 \theta - \cot^2 \theta = 1 \).