Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sec \theta \cot \theta=\csc \theta$$

Step-by-Step Solution

Answer

The trigonometric identity \( \sec \theta \cot \theta=\csc \theta \) has been verified algebraically. You can confirm it by using a graphing tool to plot both functions, which should yield matching graphs.

Step 1: Express in terms of sine and cosine

The first step is to express secant, cotangent, and cosecant functions in terms of sine and cosine. Secant is the reciprocal of cosine, cotangent is the reciprocal of tangent which is cosine divided by sine, and cosecant is the reciprocal of sine. So the identity becomes \(\frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{1}{\sin \theta}\).

Step 2: Simplify

Next, simplify the left hand side of the equation. We can simplify by cancelling the cosine in the numerator and the denominator which gives \(\frac{1}{\sin \theta}\). Now left hand side equals to the right hand side, so we have proved the identity.

Step 3: Verify graphically

To verify this result graphically, plot the graphs of \(y = \sec \theta \cot \theta\) and \(y = \csc \theta\) using a graphing utility. If the graphs overlap each other, then the identity has been verified graphically.

Key Concepts

Secant Function

The secant function, one of the six primary trigonometric functions, relates to the cosine function. It is the reciprocal of the cosine function. This means that the secant function is defined as \[ \sec \theta = \frac{1}{\cos \theta} \]. When the cosine of an angle is zero, the secant function becomes undefined as you are dividing by zero.
Understanding the secant function's behavior on the unit circle helps clarify its nature. Wherever the cosine of an angle is positive, the value of the secant is greater than 1. Conversely, when cosine is negative, the secant is less than -1. As such, the secant graph has vertical asymptotes wherever the cosine function has zeroes.
In practical applications, the secant of an angle in a right triangle can also represent the ratio of the hypotenuse to the adjacent side. This illuminates how secant connects with triangle properties and can be used in various geometric and physical problems.
It’s important to note that like the other reciprocal trigonometric functions, the secant function is periodic and repeats every \(2\pi\) radians.

Cotangent Function

The cotangent function is another reciprocal trigonometric function, which is the inverse of the tangent. It is defined as the ratio of the cosine of an angle to the sine of that angle, \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]. Cotangent, like tangent, is often associated with angle measures in right triangles, representing the ratio of the adjacent side to the opposite side.
The cotangent function shares periodic properties with tangent but differs in its domain exclusions. While tangent is undefined when the angle equals multiples of \(\pi/2\), cotangent is undefined for multiples of \(\pi\) where the sine function equals zero. This results in the cotangent graph having vertical asymptotes at these points.
Visually, when you look at the unit circle, cotangent represents how you can traverse the circle, considering the coordinates. This function provides insights into rotational dynamics and oscillatory systems, making it a useful tool in both geometric contexts and advanced fields like signal processing.
The cotangent function repeats every \(\pi\) radians, making it a crucial part of analyses involving periodic phenomena.

Cosecant Function

Cosecant is another reciprocal trigonometric function, closely related to the sine function. It is defined as the reciprocal of sine, expressed as \[ \csc \theta = \frac{1}{\sin \theta} \]. This implies that whenever the sine of an angle is zero, the cosecant becomes undefined.
In terms of practical applications, the cosecant of an angle in a right triangle can represent the ratio of the hypotenuse to the opposite side. This interpretation offers a connection between trigonometric functions and geometric reasoning, particularly within the context of circles and triangles.
Moreover, like other trigonometric functions, the cosecant function has specific periodic characteristics. It repeats every \(2\pi\) radians, similar to the sine function. By knowing where sine equals zero, you can identify that the cosecant function will have identical vertical asymptotes at those points.
Understanding the cosecant function's behavior reinforces the concept of reciprocal identities and helps unravel more complex trigonometric identities, which are significant in calculus, engineering, and physics for analyzing waveforms and harmonic motions.