Problem 25 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\tan \theta+\cos (-\theta)+\tan (-\theta)$$

Step-by-Step Solution

Answer

The simplified expression is \( \cos \theta \).

Step 1: Recognize Identities

Recall the identities: \( \tan(-\theta) = -\tan(\theta) \) and \( \cos(-\theta) = \cos(\theta) \). These identities are fundamental because tangent is an odd function while cosine is an even function.

Step 2: Substitute Identities

Apply the trigonometric identities to the expression. Replace \( \cos(-\theta) \) with \( \cos(\theta) \) and \( \tan(-\theta) \) with \(-\tan(\theta) \). The expression becomes \( \tan \theta + \cos \theta - \tan \theta \).

Step 3: Simplify the Expression

Cancel the \( \tan \theta \) and \( -\tan \theta \) terms in the expression. This leaves us with the simplified expression: \( \cos \theta \).

Key Concepts

Odd and Even Functions

Understanding odd and even functions is crucial in trigonometry. They help in simplifying expressions and solving equations efficiently. In the world of trigonometric functions:

• Odd Functions: These are functions where \( f(-x) = -f(x) \). If you apply this to trigonometric functions, the sine and tangent functions are odd. This means that \( an(-\theta) = -\tan(\theta) \), which was used in our exercise.
• Even Functions: These functions satisfy \( f(-x) = f(x) \). Cosine is an even function, indicating that \( \cos(-\theta) = \cos(\theta) \).

By knowing these properties, you can replace \( \tan(-\theta) \) and \( \cos(-\theta) \) with their respective simplified forms to transform the expression into something easier to work with. This understanding significantly reduces computational effort and complexity.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic techniques to reduce the complexity of these expressions. The goal is to express the function in its simplest form for easy computation and analysis.

• Step 1: Identify Basic Identities - Key identities like \( \tan(-\theta) = -\tan(\theta) \) and \( \cos(-\theta) = \cos(\theta) \) from even and odd function properties help set the stage for simplification.
• Step 2: Apply Transformations - Substitute the identities into the given expression. For example, replace elements of the expression based on their identities, such as turning \( \cos(-\theta) \) into \( \cos(\theta) \).
• Step 3: Cancel and Simplify - Look for terms within the expression that can be canceled or combined, simplifying the overall expression, as done with the \( \tan \theta + \tan(-\theta) = 0 \).

These steps make managing complex trigonomic functions and improving calculation efficiency more intuitive.

Tangent and Cosine Functions

The tangent and cosine functions are two important trigonometric functions that are often used together in identity simplification. Their properties allow for effective manipulation in mathematical equations.

• Tangent Function: Represented as \( \tan \theta \), it is the ratio of the sine and cosine functions. Its odd function property (\( \tan(-\theta) = -\tan(\theta) \)) is especially useful for simplifying expressions involving negative angles.
• Cosine Function: Denoted as \( \cos \theta \), it gives the horizontal coordinate on the unit circle. An even function, it satisfies \( \cos(-\theta) = \cos(\theta) \), making substitutions straightforward when dealing with negative angles.

Combining these two functions by applying their unique properties allows transformations and simplifications of expressions, such as turning complex expressions into ones that are easy to work with, as was achieved in the original solution.