Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\sin \theta \sec \theta=\tan \theta$$

Step-by-Step Solution

Answer

The identity \( \sin \theta \sec \theta = \tan \theta \) is verified.

Step 1: Express Secant in Terms of Cosine

Recall that the secant function is the reciprocal of the cosine function. Therefore, replace \( \sec \theta \) with \( \frac{1}{\cos \theta} \). The expression becomes: \( \sin \theta \cdot \frac{1}{\cos \theta} \).

Step 2: Simplify the Expression by Performing the Multiplication

Multiply \( \sin \theta \) by \( \frac{1}{\cos \theta} \) to get \( \frac{\sin \theta}{\cos \theta} \).

Step 3: Recognize the Result as Tangent

Recall that the tangent function is defined as the ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Thus, we have \( \tan \theta = \tan \theta \), which verifies the identity.

Key Concepts

Secant Function

The secant function, commonly denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions, and it is particularly useful in various mathematical contexts. Unlike the more familiar sine and cosine functions, the secant function is the reciprocal of the cosine function.

Consequently, the secant function is defined as follows:
* \( \sec \theta = \frac{1}{\cos \theta} \)

This definition means that whenever you encounter the secant function in an equation, you can express it in terms of the cosine function.

Breaking down trigonometric expressions using this reciprocal relationship allows us to transform and simplify complex trigonometric identities. This is because replacing the secant with its equivalent in terms of cosine often reveals simpler relationships between the different trigonometric terms involved.

Tangent Function

The tangent function, which is represented as \( \tan \theta \), is another essential trigonometric function that we encounter frequently in equations and identities. This function provides the ratio of the opposite side to the adjacent side in a right-angled triangle.

This relationship is also expressed in terms of sine and cosine:
* \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Understanding this direct relationship between sine, cosine, and tangent is crucial for verifying trigonometric identities. By recognizing \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \), we can often simplify and transform expressions more easily. This often involves replacing a seemingly complex tangent term with the more basic sine and cosine terms, which can then be manipulated algebraically.

Sine and Cosine Relationship

In the world of trigonometry, the sine and cosine functions are closely intertwined, and they often come up together in identities and equations. The sine function, \( \sin \theta \), and the cosine function, \( \cos \theta \), are both fundamental trigonometric functions related to angles in a unit circle.

One of the most significant properties lies in their sum of squares:
* \( \sin^2 \theta + \cos^2 \theta = 1 \)

This Pythagorean identity is one of the cornerstones of trigonometry and is always true for any angle \( \theta \). Because the tangent function is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), it relies directly on the values obtained from these two functions.

It is essential to understand these relationships because they allow you to express and manipulate complex trigonometric expressions into simpler forms, which can significantly aid in solving equations and proving identities.