Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Trigonometric Identities

First, we need to identify the relevant trigonometric identities related to the terms in the expression. We know that \( \tan(-\theta) = -\tan(\theta) \) and \( \cos(-\theta) = \cos(\theta) \). These identities will help simplify the terms in the expression.

Step 2: Apply Identities to Each Term

Applying the identities from the previous step, replace \( \tan(-\theta) \) with \( -\tan(\theta) \) and \( \cos(-\theta) \) with \( \cos(\theta) \) in the expression \( \tan \theta + \cos (-\theta) + \tan (-\theta) \) to get: \[ \tan(\theta) + \cos(\theta) - \tan(\theta) \].

Step 3: Simplify the Expression

Now, we simplify the expression by combining like terms. \( \tan(\theta) - \tan(\theta) \) cancels out to 0, leaving us with \( \cos(\theta) \). So, the expression simplifies to \( \cos(\theta) \).

Key Concepts

Simplifying Expressions

Simplifying expressions is about reducing a complex assembly of mathematical terms into a more manageable or recognizable form. With trigonometric expressions, simplifying involves manipulating variables and constants to arrive at a simpler equivalent by using basic rules or identities.

• Start by identifying and applying known identities. This often includes recognizing patterns or relationships within the expression.
• Pay attention to terms that can be combined or canceled, which substantially reduces the complexity.

In the given problem, we utilized the identity transformations to cancel terms, achieving a straightforward expression. The key is always to zero in on what terms accompany each other, especially those that can negate or support other terms. Once mastered, these strategies become invaluable in handling more intricate problems effectively.

Trigonometric Functions

Trigonometric functions, such as sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)), are foundational to understanding the relationship between angles and sides in right triangles. They form the baseline for more complex analyses in mathematics and physics.

• Tangent represents the ratio of the opposite side to the adjacent side of an angle in a right triangle. It is derived from \( \tan \theta = \rac{\sin \theta}{\cos \theta} \).
• Cosine measures the ratio of the adjacent side to the hypotenuse, \( \cos \theta = \rac{\text{adjacent}}{\text{hypotenuse}} \).

In our example, we simplified \( \tan(-\theta) \) and \( \cos(-\theta) \) to relate these functions positively to their counterparts at \( \theta \). Understanding these relationships is key to simplifying and solving many mathematical problems.

Angle Transformations

Angle transformations work on the principle that trigonometric functions exhibit specific behavior under modifications of the associated angle. For instance, the function's value might invert, retain, or mirror based on the transformation rule applied.
For negative angles:

• The tangent of a negative angle \( \tan(-\theta) \) is simply the negative of the tangent of that angle, \( -\tan(\theta) \), which emphasizes its symmetry.
• Cosine remains unaffected by sign inversion of the angle, so \( \cos(-\theta) = \cos(\theta) \).

Knowing these properties allows us to intuitively simplify expressions involving transformations. The exercise illustrated this through using identities that simplify the expression by substituting equivalent terms. By mastering angle transformations, not only do problems become easier, but also we gain a spatial understanding of how angles interact within mathematical space.