Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\cot \left(\frac{\pi}{2}-x\right) \cos x$$

Step-by-Step Solution

Answer

The simplified expression is \( \tan(x) \times \cos(x)\)

Step 1: Identify the trigonometric identities

We begin by identifying the relevant trigonometric identities. For the cotangent function, \(\cot(a) = \frac{1}{\tan(a)}\). For the cosine function, \(\cos(a) = \sin(\frac{\pi}{2} - a)\).

Step 2: Substitute the identities into the expression

Substituting these identities into the expression yields: \(\frac{1}{\tan(\frac{\pi}{2}-x)} \times \sin(\frac{\pi}{2} - x)\). We further simplify by noting that \(\tan(\frac{\pi}{2} - x) = \frac{1}{\tan(x)}\). Substituting this into our expression, we get: \(\frac{1}{\frac{1}{\tan(x)}} \times \sin(\frac{\pi}{2} - x)\).

Step 3: Simplify the expression

This simplifies to \( \tan(x) \times \sin(\frac{\pi}{2} - x)\). From trigonometric identity, we know that \(\sin(\frac{\pi}{2} - x) = \cos(x)\). Substituting this gives us the final simplified expression: \( \tan(x) \times \cos(x)\)

Key Concepts

Fundamental Trigonometric Identities

The world of trigonometry is built upon a framework of core principles known as the fundamental trigonometric identities. These identities are equalities that involve trigonometric functions and hold true for all values of the variables within their domain of definition. Among the most essential ones are the Pythagorean identities, reciprocal identities, and angle sum and difference identities.

For instance, the reciprocal identities relate the primary trigonometric functions to their 'co-' functions, such as \(\cos(x) = \frac{1}{\sec(x)}\) and \(\sin(x) = \frac{1}{\csc(x)}\). Understanding these identities provides a solid foundation for manipulating and simplifying complex trigonometric expressions, making it essential for students to commit them to memory and know how to apply them in various scenarios.

Cotangent Function

The cotangent function, denoted as \(\cot\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, meaning \(\cot(x) = \frac{1}{\tan(x)}\) or equivalently \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Importantly, the cotangent function exhibits periodic behavior, inherent to trigonometric functions, with a period of \(\pi\) radians, or 180 degrees.

Understanding the cotangent function's graphical representation can be challenging. Its graph exhibits discontinuities wherever the sine function equals zero, as the cotangent function is undefined when \(\sin(x) = 0\). Mastery of the cotangent function, including its properties and behavior, is vital in simplifying trigonometric expressions involving this function.

Graphing Utility

In the modern era, graphing utilities have become an indispensable tool for visualizing mathematical concepts and verifying solutions. They serve as a practical application of technology in education, allowing students to see the behavior of trigonometric functions graphically. A graphing utility can plot the shapes of these functions based on their fundamental properties, like amplitude, period, and phase shift.

Students can input an expression such as \(\cot(\frac{\pi}{2}-x)\cos(x)\) into a graphing utility to verify the result obtained through algebraic simplification. By using the table feature, for example, they can compare numerical values of the original and simplified expressions, thus reinforcing their understanding of trigonometric identities and their ability to simplify complex expressions.

Trigonometric Simplification Process

The process of trigonometric simplification involves reducing complex trigonometric expressions to simpler forms, often making use of the fundamental trigonometric identities. This can involve transforming products into sums, applying Pythagorean identities, or utilizing angle sum and difference formulas.

The simplification process typically follows a logical sequence: identify applicable identities, substitute the identities into the expression, and algebraically manipulate the result into a simplified form. For example, \(\cot(\frac{\pi}{2}-x)\cos(x)\) can be simplified by recognizing that \(\sin(\frac{\pi}{2} - x) = \cos(x)\) and \(\tan(\frac{\pi}{2} - x) = \cot(x)\), thus leading to the simplified expression \(\tan(x)\cos(x)\). Knowing when and how to apply specific identities is a key skill in trigonometric simplification, enhancing problem-solving efficiency and ensuring accuracy.