Problem 17 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sec \theta \cos \theta \sin \theta $$

Step-by-Step Solution

Answer

The simplified form of \( \sec \theta \cos \theta \sin \theta \) is \( \sin \theta \).

Step 1: Understand the Relationship between Secant and Cosine

Recall the definition of secant in terms of cosine, which is \( \sec \theta = \frac{1}{\cos \theta} \). So, wherever \( \sec \theta \) is found, it could be replaced with \( \frac{1}{\cos \theta} \).

Step 2: Substitute Secant in the Expression

Replace \( \sec \theta \) in the original expression \( \sec \theta \cos \theta \sin \theta \) with \( \frac{1}{\cos \theta} \). So, we get \( \frac{1}{\cos \theta} \cos \theta \sin \theta \).

Step 3: Simplify the Expression

Now, there's \( \cos \theta \) in the numerator and denominator. They will cancel out each other, which leaves just \( \sin \theta \).

Key Concepts

Secant Function

The secant function, often abbreviated as "sec," is one of the six main trigonometric functions. It is considered the reciprocal of the cosine function. This means that the secant of an angle \( \theta \) can be expressed as:

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

Whenever you're dealing with an expression involving the secant function, it's often useful to look at it in terms of cosine. This is because many trigonometric expressions and identities are easier to simplify or evaluate when everything is expressed in terms of sine and cosine. By understanding the reciprocal nature here, we directly impact how we simplify expressions like \( \sec \theta \cos \theta \sin \theta \). You'll often find that converting \( \sec \theta \) to \( \frac{1}{\cos \theta} \) is the first step when handling such problems.Trigonometry involves looking at triangles and their relationships, and the secant function, just like its counterparts, helps us understand these relationships better. The beauty of trigonometry is its interconnectedness, where knowing one function can help deduce others, and simplify complex looking expressions.

Cosine Function

The cosine function is a fundamental component in trigonometry, often symbolized as "cos." It is one of the primary functions used to relate angles to the sides of a right triangle. Specifically, for an angle \( \theta \), the cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse. It can be written as:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

The cosine function is versatile and used in various applications, from simple angle calculations to more complex waveform analysis in physics and engineering.Through trigonometric identities, we see that the cosine function complements the sine function. Its role becomes particularly important when simplifying expressions like \( \sec \theta \cos \theta \sin \theta \); as demonstrated, knowing \( \cos \theta \) can help cancel terms when paired with its reciprocal. The cosine function serves as a bridge for transforming and simplifying expressions, and its integration with the secant function becomes key for solving complicated trigonometric equations.

Sine Function

The sine function, or "sin," is equally a cornerstone of trigonometry as the cosine function. It relates the angle \( \theta \) of a right triangle to the ratio of the side opposite the angle to the hypotenuse. The sine function is defined by the following ratio:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

This function is crucial in various mathematical and real-life applications, such as modeling periodic phenomena like sound waves and light waves.In the context of trigonometric simplification, the sine function often plays a role in the final form of the solution. For instance, in the expression \( \sec \theta \cos \theta \sin \theta \), after breaking down secant in terms of cosine and simplifying, \( \sin \theta \) is all that remains. Thus, understanding sine is not just about knowing its definition but recognizing how it appears and operates within broader trigonometric expressions. Simplifying and interpreting trigonometric equations often leads to seeing the sine function as the end result.