Problem 136 - Precalculus with Limits

Question

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer. \( \dfrac{1}{\left(5 \cos \theta\right)} = 5 \sec \theta \)

Step-by-Step Solution

Answer

Yes, the given equation \( \dfrac{1}{\left(5 \cos \theta\right)} = 5 \sec \theta \) is an identity.

Step 1: Identify the secant

Begin by recognizing secant as the reciprocal of cosine, i.e., \( \sec \theta = \dfrac{1}{\cos \theta} \)

Step 2: Substitute secant with its equivalent

Replace \(\sec \theta\) in the original equation with \( \dfrac{1}{\cos \theta}\) which gives \( \dfrac{1}{\left(5\cos \theta\right)} = 5 \left(\dfrac{1}{\cos \theta}\right) \)

Step 3: Simplify RHS

Simplify the right-hand side (RHS) of the equation to have \( \dfrac{1}{\left(5\cos \theta\right)} = \dfrac{5}{\cos \theta} \)

Step 4: Compare LHS and RHS

Now the left-hand side (LHS) and RHS of the equation look the same, hence verifying that the given equation is an identity. Therefore, \( \dfrac{1}{\left(5\cos \theta\right)} = 5 \sec \theta \) is an identity because both sides are equal for all values of \( \theta \) for which cos(\( \theta \)) is defined and non-zero.

Key Concepts

Secant Function

The secant function is a fascinating part of trigonometry, providing a unique perspective on angles and their relationships. It is one of the six primary trigonometric functions, less commonly used compared to sine and cosine, but equally important. The secant function, denoted as \( \sec \theta \), is defined as the reciprocal of the cosine function. In mathematical terms, it is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This means that for any angle \( \theta \), the secant is equal to one divided by the cosine of that angle.
This relationship makes it clear why secant is undefined whenever the cosine of \( \theta \) is zero; division by zero is undefined. Therefore, \( \sec \theta \) is undefined at angles like \( 90^\circ \) and \( 270^\circ \), where \( \cos \theta \) equals zero. The secant function, along with its reciprocal nature, plays a crucial role in simplifying and solving trigonometric equations.

Cosine Function

The cosine function is one of the foundational building blocks of trigonometry. It is typically introduced in the context of right triangles but extends to applications in circular functions and waves. In a right triangle, the cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse:

• \( \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

This gives us a geometric perspective on the cosine function. Additionally, on the unit circle, where the radius (hypotenuse) is one, the cosine of an angle corresponds directly to the x-coordinate of the point on the circle.
Cosine remains a critical function, useful in describing oscillations such as sound waves or electrical currents, and in many areas of physics and engineering. It also seamlessly interconnects with other trigonometric functions, highlighting its versatility and importance.

Reciprocal Identities

Reciprocal identities in trigonometry provide powerful tools for transforming and simplifying expressions and equations. These identities express each trigonometric function in terms of its reciprocal. For example, the reciprocal identities for basic functions are:

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

These identities are invaluable when solving or verifying equations, such as evaluating whether an equation is an identity. By expressing complex trigonometric equations in terms of their reciprocal identities, they often become simpler and more manageable.
In our original exercise, the secant function was rewritten using its reciprocal identity, simplifying the equation and highlighting the fundamental relationship between secant and cosine. The consistent use of reciprocal identities can lead to greater computational ease and clarity in trigonometric problem-solving.