Problem 25 - Algebra 2

Question

Simplify each trigonometric expression. $$ \cos ^{2} \theta \sec \theta \csc \theta $$

Step-by-Step Solution

Answer

The simplified form of the trigonometric expression is \(\cot\theta\).

Step 1: Understand the given expression

The given expression is \( \cos^{2}\theta \cdot \sec\theta \cdot \csc\theta \). Here, \(\sec\theta\) is the reciprocal of \(\cos\theta\) and \(\csc\theta\) is the reciprocal of \(\sin\theta\). Knowing these relationships is crucial to simplifying the expression.

Step 2: Replace with equivalent identities

Replace \(\sec\theta\) with \(1/\cos\theta\) and replace \(\csc\theta\) with \(1/\sin\theta\). Doing so, the expression becomes \( \cos^{2}\theta \cdot (1/\cos\theta) \cdot (1/\sin\theta)\)

Step 3: Simplify

Now, simplify the new expression. \( \cos\theta \cdot (1/\sin\theta) = \cot\theta\) because \(\cot\theta\) is the ratio of \(\cos\theta\) to \(\sin\theta\). So, the simplified expression is \(\cot\theta\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in studying angles and their relationships. These functions include sine (\(\sin \theta\)), cosine (\(\cos \theta\)), tangent (\(\tan \theta\)), and their reciprocals. Understanding these functions is crucial as they describe the relationship between angles and sides in a right triangle.

• Sine (\(\sin \theta\)): Represents the ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos \theta\)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan \theta\)): Represents the ratio of the opposite side to the adjacent side.

Trigonometric functions have numerous applications in various fields such as physics, engineering, and architecture. They help in modeling periodic phenomena like sound and light waves. It is important to grasp these concepts as they form the foundation for understanding more complex mathematical concepts.

Simplifying Expressions

Simplifying trigonometric expressions involves reducing them to their simplest form while maintaining equality. This process often uses identities and algebraic manipulation to achieve a more basic representation of the expression. Simplifying can make complex equations easier to handle and understand.

• Simplification helps in revealing underlying patterns or behaviors in trigonometric expressions.
• It often involves combining like terms and using identities to replace more complex terms.

When simplifying expressions:

• Identify the trigonometric identities that can be applied.
• Systematically simplify each part of the expression.
• Ensure that each step maintains the equality of the expression.

Proper understanding and practice in simplifying expressions can lead to more efficient problem-solving and can significantly clarify complex mathematical tasks.

Reciprocal Identities

Reciprocal identities are key trigonometric identities used to simplify expressions. These identities express a trigonometric function in terms of its reciprocal. The main reciprocal identities include:

• Secant (\(\sec \theta\)): The reciprocal of cosine (\(\cos \theta\)), which is \(\sec\theta = \frac{1}{\cos\theta}\);
• Cosecant (\(\csc \theta\)): The reciprocal of sine (\(\sin \theta\)), which is \(\csc\theta = \frac{1}{\sin\theta}\);
• Cotangent (\(\cot \theta\)): The reciprocal of tangent (\(\tan \theta\)), which is \(\cot\theta = \frac{1}{\tan\theta}\).

In the context of our exercise, recognizing and applying reciprocal identities was crucial to simplifying the given trigonometric expression. By substituting \(\sec\theta\) with \(\frac{1}{\cos\theta}\) and \(\csc\theta\) with \(\frac{1}{\sin\theta}\), we broke down a more complex expression into simpler terms. Understanding how and when to use these identities can dramatically ease complexity in mathematical problems.