Problem 39 - Basic Technical Mathematics with Calculus

Question

Answer the given questions. From the definitions of the trigonometric functions, it can be seen that \(\csc \theta\) is the reciprocal of \(\sin \theta\). What function is the reciprocal of \(\cos \theta ?\)

Step-by-Step Solution

Answer

The reciprocal of \( \cos \theta \) is \( \sec \theta \).

Step 1: Understanding the Problem

The problem is asking for the reciprocal of the trigonometric function \( \cos \theta \). We need to identify which trigonometric function corresponds to this reciprocal relationship.

Step 2: Defining Reciprocal Functions

A reciprocal function in trigonometry means that if \( f(\theta) \) is a trigonometric function, its reciprocal is \( \frac{1}{f(\theta)} \). For example, \( \csc \theta = \frac{1}{\sin \theta} \), showing \( \csc \theta \) is the reciprocal of \( \sin \theta \).

Step 3: Identifying the Reciprocal of \( \cos \theta \)

To find the reciprocal of \( \cos \theta \), we divide 1 by \( \cos \theta \). The function that represents \( \frac{1}{\cos \theta} \) is known as \( \sec \theta \).

Step 4: Conclusion

The trigonometric function that serves as the reciprocal of \( \cos \theta \) is \( \sec \theta \).

Key Concepts

Reciprocal Functions

Reciprocal functions in trigonometry work similarly to how reciprocals function in basic arithmetic. In arithmetic, the reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 5 is \( \frac{1}{5} \).In trigonometry, each primary trigonometric function has a corresponding reciprocal function. These functions are:

• Sine (\( \sin \theta \)) whose reciprocal is cosecant (\( \csc \theta \))
• Cosine (\( \cos \theta \)) whose reciprocal is secant (\( \sec \theta \))
• Tangent (\( \tan \theta \)) whose reciprocal is cotangent (\( \cot \theta \))

This relationship means that if you know any basic trigonometric function, you can easily find its reciprocal by taking one over that function. The concept of reciprocal functions is crucial because it allows for the derivation of useful identities and simplifications in trigonometry, particularly in the context of solving equations involving trigonometric expressions.

Cosecant

Cosecant, represented as \( \csc \theta \), is the reciprocal of the sine function. When dealing with angles and their trigonometric functions in a right triangle, remember: - If \( \sin \theta \) describes the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse, then \( \csc \theta \) is \( \frac{1}{\sin \theta} \). - Essentially, if you have calculated \( \sin \theta \), finding \( \csc \theta \) simply involves taking the reciprocal of that value, i.e., \( \csc \theta = \frac{1}{\sin \theta} \).Cosecant is particularly useful in scenarios where you have the opposite side and need to determine the hypotenuse. It's important to realize that \( \csc \theta \) will always be undefined for angles \( \theta \) where \( \sin \theta = 0 \). This is because dividing by zero is undefined, so we generally avoid these angles in practice.

Secant

Secant, denoted as \( \sec \theta \), is the reciprocal of the cosine function \( \cos \theta \). It represents the ratio of the length of the hypotenuse to the adjacent side in a right triangle.- To calculate \( \sec \theta \), take one divided by \( \cos \theta \), giving: \( \sec \theta = \frac{1}{\cos \theta} \).- If \( \cos \theta \) is the ratio that pitches the adjacent side of a right triangle over the hypotenuse, \( \sec \theta \) is essentially acting in reverse to provide the ratio of the hypotenuse over the adjacent side.Secant is undefined for angles \( \theta \) where \( \cos \theta = 0 \), which typically occurs when \( \theta \) equals odd multiples of \( \frac{\pi}{2} \) radians (or 90 degrees in degrees). Understanding where these singularities occur helps prevent common computational errors in trigonometry, especially in calculating angles or interpreting trigonometric models.