Problem 43 - Precalculus

Question

Evaluate each expression, if possible. $$\csc \left(-\frac{7 \pi}{2}\right)-\cot \left(\frac{7 \pi}{2}\right)$$

Step-by-Step Solution

Answer

The expression is undefined due to the undefined cotangent term at \(\frac{\pi}{2}\).

Step 1: Understanding the Given Expression

We need to evaluate the expression \(\csc \left(-\frac{7 \pi}{2}\right) - \cot \left(\frac{7 \pi}{2}\right)\). Recall that the cosecant function \(\csc\) is the reciprocal of the sine function and the cotangent function \(\cot\) is the reciprocal of the tangent function. Both functions are periodic, which will help simplify the calculation.

Step 2: Simplifying Angles using Periodicity

The periodicity of \(\csc(\theta)\) and \(\cot(\theta)\) are based on the sine and tangent functions, which have periods of \(2\pi\) and \(\pi\) respectively. Simplifying the angles given in terms of these periods can make calculations easier: - For \(-\frac{7\pi}{2}\) in \(\csc\left(x\right)\), note \(-\frac{7\pi}{2} + 3(2\pi) = \frac{\pi}{2}\). - For \(\frac{7\pi}{2}\) in \(\cot(x)\), note \(\frac{7\pi}{2} - 3(2\pi) = \frac{\pi}{2}\).

Step 3: Calculating the Cosecant

Now, compute \(\csc\left(\frac{\pi}{2}\right)\). The sine of \(\frac{\pi}{2}\) is 1; hence, its reciprocal is also 1. Therefore, \(\csc\left(\frac{\pi}{2}\right) = 1\).

Step 4: Calculating the Cotangent

Next, calculate \(\cot\left(\frac{\pi}{2}\right)\). Since \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and \(\cos\left(\frac{\pi}{2}\right) = 0\), \(\cot\left(\frac{\pi}{2}\right)\) is undefined as it results in division by zero.

Step 5: Evaluating the Final Expression

Now substitute back into the original expression: \(\csc\left(-\frac{7\pi}{2}\right) - \cot\left(\frac{7\pi}{2}\right) = 1 - \text{undefined}\). As one of the terms is undefined, the expression as a whole cannot be evaluated into a real number.

Key Concepts

Cosecant Function

The cosecant function, often written as \( \csc(x) \), is a trigonometric function that is the reciprocal of the sine function. If you know the value of \( \sin(x) \), you can find \( \csc(x) \) by simply taking the reciprocal, meaning \( \csc(x) = 1 / \sin(x) \). This means that wherever the sine function is zero, the cosecant function is undefined, because division by zero is not allowed in mathematics.

This function is useful for problems where you need to find lengths of sides in triangles, especially when dealing with right triangles. It helps to remember key values of sine to find cosecant easily. For example, at \( \pi/2 \), \( \sin(\pi/2) = 1 \) so \( \csc(\pi/2) = 1 \) as well.

• The graph of the cosecant function has vertical asymptotes where the sine function is zero.
• It has a repeated pattern or \'period\', just like other trigonometric functions.

Recognizing these characteristics helps in solving trigonometric problems efficiently.

Cotangent Function

Another essential trigonometric function is the cotangent. Represented as \( \cot(x) \), it is the reciprocal of the tangent function. Specifically, it is defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). This means if either the sine is zero, the cotangent will be undefined due to division by zero.

In trigonometry, the cotangent function is useful in certain types of angle calculations, especially in right triangles. For certain angles, such as \( \pi/2 \), the cosine is zero, leading to the cotangent being undefined.

• The function, like other trigonometric functions, has its distinct pattern of repetition or periodicity.
• It alternates between positive and negative values as you move along the angle axis because of the varied signs of the sine and cosine in different quadrants.

Understanding the behavior and limitations of cotangent can help you effectively evaluate and simplify trigonometric expressions.

Periodicity

Periodicity is an important concept in trigonometry where the functions repeat their values at regular intervals. This is especially true for the basic trigonometric functions like sine, cosine, and their reciprocals—cosecant and cotangent.

For the cosecant and sine functions, this period is \(2\pi\) because after this interval, the function values start to repeat. On the other hand, the cotangent function, along with the tangent, has a shorter period of \(\pi\).

• Using periodicity can simplify trigonometric calculations. For example, \(\csc(-\frac{7\pi}{2})\) simplifies to \(\csc(\frac{\pi}{2})\) after removing multiples of \(2\pi\).
• It also means you do not always need the exact angle, but only how far it is from any period boundary.

Recognizing this pattern allows you to evaluate expressions that might seem complex at first glance, using simpler angles.

Undefined Expression

An undefined expression often arises in mathematics when the evaluation leads to a division by zero. In trigonometry, this can occur with functions like cotangent and cosecant where the sine is the denominator. If \( \sin(x) = 0 \), then both \( \csc(x) \) and \( \cot(x) \) become undefined.

For the original exercise, \( \cot(\frac{\pi}{2}) \) results in an undefined expression because \( \sin(\frac{\pi}{2}) = 0 \), leading to division by zero.

• These undefined expressions require careful handling and awareness.
• They often signify vertical asymptotes on their respective graphs.

Identifying these points is crucial as they can determine whether an expression can actually be evaluated into a finite number or not. It's important to note these when solving mathematical problems.