Problem 13 - Technical Mathematics with Calculus

Question

Simplify. $$\sec x \sin x$$

Step-by-Step Solution

Answer

\(\sec x \sin x = \tan x\)

Step 1: Understanding the trigonometric identities

To simplify the expression \(\sec x \sin x\), we need to recall that \(\sec x\) is the reciprocal of \(\cos x\), so \(\sec x = \frac{1}{\cos x}\).

Step 2: Applying the reciprocal identity

Replace \(\sec x\) with \(\frac{1}{\cos x}\) to get the expression \(\frac{1}{\cos x} \sin x\).

Step 3: Multiplying the trigonometric functions

Multiply the sine function by the reciprocal of the cosine function, which gives us \(\frac{\sin x}{\cos x}\).

Step 4: Using the quotient identity

Recognize that \(\frac{\sin x}{\cos x}\) is the definition of the tangent function, so the expression simplifies to \(\tan x\).

Key Concepts

Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are fundamental in simplifying complex trigonometric expressions. They involve the three basic reciprocal pairs: sine and cosecant, cosine and secant, and tangent and cotangent. In the expression \( \sec x \sin x \), the secant function, \( \sec x \), is the reciprocal of the cosine function, represented as \( \sec x = \frac{1}{\cos x} \). Likewise, \( \csc x \) is the reciprocal of \( \sin x \), and \( \cot x \) is the reciprocal of \( \tan x \).

Understanding these identities allows you to transform expressions to more familiar terms and often to simplify them drastically. When simplifying trigonometric expressions using reciprocal identities, you should look for opportunities to replace functions for their reciprocals, particularly when functions are being multiplied together, as this can lead to a direct simplification or even cancel out terms.

Sine and Cosine Functions

The sine and cosine functions are two of the most important trigonometric functions, deeply intertwined and foundational in the study of triangles and circles. The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse. Similarly, the cosine is the ratio of the adjacent side to the hypotenuse.

In the context of the unit circle, where all radii are of length 1, these ratios can be understood as the y-coordinate (sine) and x-coordinate (cosine) of a point on the circle's circumference corresponding to a given angle. These functions are periodic, with sine and cosine completing a full cycle every \(2\pi\) radians or 360 degrees.

When working with products of sine and cosine functions, as in \(\sec x \sin x\), you have to consider their relationship to each other. In our example, we are essentially multiplying \(\sin x\) with \(1/\cos x\), and this understanding is crucial for the simplification process.

Tangent Function

The tangent function is another primary trigonometric function that can be understood in terms of sine and cosine. Tangent is defined as the ratio of the sine of an angle to the cosine of that same angle, mathematically expressed as \(\tan x = \frac{\sin x}{\cos x}\).

The key to recognizing when to use this identity comes from observing ratios of sine to cosine within an expression. In our exercise, after applying the reciprocal identity of secant, we encountered the fraction \(\frac{\sin x}{\cos x}\), which is the definition of tangent. This is a very powerful observation, enabling the simplification of \(\sec x \sin x\) to simply \(\tan x\).

Therefore, understanding how to convert between \(\sin x\), \(\cos x\), and \(\tan x\) through these trigonometric identities can greatly simplify many trigonometry problems. Remembering the basic definitions of these functions and their relationships will help you simplify expressions and solve trigonometric equations more efficiently.