Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression so that it is not in fractional form. $$\frac{\csc y}{\cot y}$$

Step-by-Step Solution

Answer

\(\sec y\) is the simplified form of the given expression

Step 1: Identify Definitions

The first step is to recognise the functions involved. The cosecant function (\(\csc y\)) is defined as \(1/\sin y\) and the cotangent function (\(\cot y\)) is defined as \(\cos y/\sin y\).

Step 2: Substitute Values

Now substitute these values into the equation giving \(\frac{1/\sin y}{\cos y/\sin y}\).

Step 3: Simplify

The division of fractions is equivalent to the multiplication of the first fraction by the reciprocal of the second. In other words, if you have \(a/b\) divided by \(c/d\), it becomes \(a/b\) times \(d/c\). Hence \(\frac{1/\sin y}{\cos y/\sin y}\) becomes \((1/\sin y)\times(\sin y/\cos y)\).

Step 4: Further Simplify

Now simplify the expression. As you will notice, \(\sin y\) in the numerator and denominator of the expression will cancel each other out and we are left with \(1/\cos y\).

Step 5: Return to Original Function

Finally, in trigonometry, \(1/\cos y\) is equivalent to the secant function (\(\sec y\)). Hence, the expression simplifies to \(\sec y\).

Key Concepts

Cosecant Function

The cosecant function is one of the six fundamental trigonometric functions. It's particularly useful when dealing with right-angled triangles. The cosecant, often abbreviated as \( \csc \), is the reciprocal of the sine function. Therefore, for an angle \( y \), cosecant is defined as:

• \( \csc y = \frac{1}{\sin y} \)

This function is undefined when \( \sin y = 0 \) because division by zero is not possible. Such points coincide with the values where \( y \) is a multiple of \( \pi \) (or 180 degrees). Understanding \( \csc y \) is crucial in solving equations or identities that involve inverse relationships to the sine function.

In practical terms, if you know the sine of an angle, you can easily find the cosecant by taking its reciprocal. This is very helpful in simplifying expressions and solving trigonometric equations.

Cotangent Function

The cotangent function is another key trigonometric function, often abbreviated as \( \cot \). It's the reciprocal of the tangent function. For an angle \( y \), cotangent is given by the formula:

• \( \cot y = \frac{1}{\tan y} \)
• Equivalently, \( \cot y = \frac{\cos y}{\sin y} \)

This definition indicates that the cotangent function is undefined when \( \sin y = 0 \), which occurs at multiples of \( \pi \). Its undefined angles correspond directly with those of the cosecant function, due to their shared dependence on \( \sin y \) in the denominator. The cotangent function is extremely useful in various applications like simplifying trigonometric expressions or integrating functions. It helps in reversing a tangent value, which is a common requirement when solving trigonometric equations.

Secant Function

The secant function, denoted as \( \sec \), is the reciprocal of the cosine function. For an angle \( y \), it can be defined as:

• \( \sec y = \frac{1}{\cos y} \)

This makes \( \sec y \) undefined wherever \( \cos y = 0 \), those are the odd multiples of \( \frac{\pi}{2} \). These are the points at which the cosine function crosses the horizontal axis. Understanding secant is essential for graphing and transforming trigonometric functions.

In the context of the exercise given, we started with \( \frac{\csc y}{\cot y} \). By substituting their definitions and simplifying, this expression transformed into \( \sec y \). This demonstrates how interconnected these trigonometric identities are and how they can simplify seemingly complex fractional expressions.