Problem 51 - Precalculus: Functions and Graphs

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 is verified: \( \sin \theta \sec \theta = \tan \theta \).

Step 1: Substitute Basic Trigonometric Identities

To verify the identity \( \sin \theta \sec \theta = \tan \theta \), we'll substitute the basic identities. Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this identity into the left-hand side: \( \sin \theta \cdot \frac{1}{\cos \theta} \).

Step 2: Simplify the Expression

Continuing from Step 1, the expression becomes \( \frac{\sin \theta}{\cos \theta} \).

Step 3: Recognize the Result as a Trigonometric Function

Recall that \( \frac{\sin \theta}{\cos \theta} \) is the identity for \( \tan \theta \). Therefore, \( \sin \theta \sec \theta = \tan \theta \) as required by the problem statement.

Key Concepts

Understanding the Sin Function

The sin function, or sine function, is a fundamental component of trigonometry, often found in various mathematical problems related to angles and triangles. This function is typically written as \( \sin \theta \), where \( \theta \) represents an angle in either degrees or radians. The sin function is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

• For a right triangle, \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• The value of \( \sin \theta \) ranges from -1 to 1.
• It is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

Knowing the properties of the sin function can help in understanding how it relates to other trigonometric identities, like the one used in transforming it to \( \tan \theta \) in this problem. By understanding sine, we can express angles and their magnitudes in various forms, aiding in simplifying complex expressions.

The Role of the Sec Function

The sec function, also known as secant, is another trigonometric function that serves as the reciprocal of the cosine function. It is expressed as \( \sec \theta \), defined mathematically as:

• \( \sec \theta = \frac{1}{\cos \theta} \).

This function is essential because it allows us to express certain trigonometric identities, as seen in the given exercise, where it helps transform \( \sin \theta \sec \theta \) into \( \tan \theta \).

Key Points to Remember:

• Since it is the reciprocal of the cosine, \( \sec \theta \) is undefined wherever \( \cos \theta = 0 \).
• It is useful in simplifying expressions and solving equations involving trig identities.
• Like other trigonometric functions, it is periodic with a period of \(2\pi\).

Understanding secant helps break down complex trigonometric problems into manageable steps, facilitating easier manipulation and transformation of identities.

Exploring the Tan Function

The tan function, or tangent, is a trigonometric function that connects sine and cosine through their ratios. In the context of the problem, understanding tangent is pivotal since the solution transforms other functions into \( \tan \theta \). The tangent is defined as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

This relationship is precisely how we deduced that \( \sin \theta \sec \theta \) is equal to \( \tan \theta \) after substitution and simplification.

Why is Tangent Important?

• It represents the slope of the angle when plotted on the unit circle.
• Tangent is periodic with a period of \(\pi\), repeating more frequently than sine or cosine.
• Knowing tangent values helps in solving equations related to angles and is crucial in calculus and physics.

Understanding the tan function is key to recognizing its presence in expressions and equations, like the transformation from \( \sin \theta \cdot \sec \theta \) to \( \tan \theta \) in this exercise, allowing for smooth verification and simplification.