Problem 248 - Precalculus

Question

For the following exercises, use identities to simplify the expression. csc t tan \(t\)

Step-by-Step Solution

Answer

\( \csc t \tan t = \sec t \).

Step 1: Recall Trigonometric Identities

To simplify the expression \( \csc t \tan t \), recall the definitions of the trigonometric functions involved: \( \csc t = \frac{1}{\sin t} \) and \( \tan t = \frac{\sin t}{\cos t} \).

Step 2: Substitute Definitions

Replace \( \csc t \) and \( \tan t \) in the expression with their identities: \( \csc t \tan t = \frac{1}{\sin t} \times \frac{\sin t}{\cos t} \).

Step 3: Simplify the Expression

In the expression \( \frac{1}{\sin t} \times \frac{\sin t}{\cos t} \), the \( \sin t \) in the numerator and the \( \sin t \) in the denominator cancel each other out, leaving \( \frac{1}{\cos t} \).

Step 4: Identify the Final Expression

The final simplified form of the expression \( \frac{1}{\cos t} \) is another trigonometric identity, \( \sec t \). Thus, \( \csc t \tan t = \sec t \).

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as csc, is one of the reciprocal trigonometric functions. It is related to the sine function, and is defined as the reciprocal of the sine of an angle. This means that:

• \( \csc t = \frac{1}{\sin t} \)

This function is useful in trigonometry for solving equations and simplifying expressions that involve angles. Since the sine function gives the vertical coordinate of a point on the unit circle, the cosecant function can be seen as giving the reciprocal of this measurement.
When using the cosecant function, remember its domain excludes angles where the sine equals zero, such as \(t = 0, \pi, 2\pi, \ldots\). This is because division by zero is undefined. Therefore, understanding where the sine function equals zero helps you use and understand the cosecant function effectively in various problems.

Tangent Function

The tangent function, symbolized as tan, is another fundamental trigonometric function. It represents the ratio of the sine and cosine of the same angle. Mathematically, it is expressed as:

• \( \tan t = \frac{\sin t}{\cos t} \)

The tangent function is essential in trigonometry, particularly in calculating angles and distances. It is derived from right-angled triangles, where it measures the slope or steepness of an angle relative to the x-axis.
A key point about the tangent function is its periodicity. It repeats every \(\pi\) radians, which means its graph will exhibit a pattern that repeats at these intervals. The tangent function has vertical asymptotes where the cosine function equals zero, such as \(t = \frac{\pi}{2} + n\pi\), where n is an integer. This understanding helps visualize and simplify problems involving tangents.

Secant Function

The secant function, denoted as sec, is the reciprocal of the cosine function. It is defined as the reciprocal of the cosine of an angle, which can be written as:

• \( \sec t = \frac{1}{\cos t} \)

The secant function provides valuable insights when working with trigonometric identities and simplifying complex expressions. In terms of the unit circle, the secant function represents the reciprocal of the x-coordinate of a point on the circle.
Notably, the secant function is not defined for angles where the cosine is zero, such as \(t = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots\). At these points, the secant approaches infinity, leading to vertical asymptotes in its graph.
Understanding the behavior and properties of the secant function is crucial for solving trigonometric equations and verifying identities, such as in the provided exercise where \( \csc t \tan t = \sec t \). This simplification relies on recognizing that the resulting expression, \( \frac{1}{\cos t} \), directly corresponds to the secant.