Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each expression as a single term using \(\sin \theta, \cos \theta,\) or both. $$ \tan \theta \sec \theta \cot \theta $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sec^2 \theta \).

Step 1: Write in terms of sine and cosine

First, express each trigonometric function in terms of \( \sin \theta \) and \( \cos \theta \). We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 2: Substitute the expressions

Substitute the expressions from Step 1 into the given expression. We have: \[ \tan \theta \sec \theta \cot \theta = \frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta}. \]

Step 3: Simplify the expression

Cancel out the terms in the numerator and the denominator. The \( \cos \theta \) in the numerator and denominator will cancel each other, as will the \( \sin \theta \), leaving us with \[ \frac{1}{\cos^2 \theta}. \]

Step 4: Write in terms of secant

Recognize that \( \frac{1}{\cos^2 \theta} \) is \( \sec^2 \theta \). Thus, the expression simplifies to \( \sec^2 \theta \).

Key Concepts

Sine and Cosine

Trigonometry often begins with understanding sine and cosine, the foundational trigonometric functions. These functions relate the angles and sides of a right triangle.

• The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the side opposite angle \( \theta \) to the hypotenuse of the triangle.
• The cosine function, denoted as \( \cos \theta \), represents the ratio of the length of the adjacent side to the hypotenuse.

A right triangle helps visualize these ratios, but they are defined for all angles, using the unit circle.
The unit circle is a circle centered at the origin of a coordinate plane with a radius of 1. It allows us to extend the definitions of sine and cosine to angles greater than 90 degrees and to negative angles.
In our exercise, expressing each trigonometric function in terms of \( \sin \theta \) and \( \cos \theta \) helps in simplifying the expression, as these are the basic and most easily manageable forms.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where they are defined. They are tools that simplify and solve trigonometric expressions and equations.

• The reciprocal identities, which allow us to express secant, cosecant, and cotangent in terms of sine and cosine, are crucial. In the exercise, we used the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• The Pythagorean identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \), help in simplifying and transforming expressions further if needed.

Trigonometric identities make complex expressions more manageable by breaking them down into more fundamental components, leading to straightforward simplifications.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a simpler or more efficient form without changing the value. This often includes using trigonometric identities to eliminate terms and combine like terms.

• In the resolved exercise, we started by rewriting each trigonometric function using sine and cosine. This step is essential because the basic trigonometric functions are simpler to work with and provide a common base for manipulation.
• Next, we substituted these expressions into the original, leading to an expression that allows for straightforward simplification: \( \frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \).
• Upon simplification, where terms cancel out, the expression results in \( \sec^2 \theta \). Recognizing \( \frac{1}{\cos^2 \theta} \) as \( \sec^2 \theta \) highlights the use of trigonometric identites to achieve this simplification.

Successful simplification often involves proceeding step-by-step, leveraging known identities, and addressing each component carefully, which makes seemingly complex problems more understandable and solvable.