Problem 24 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sin ^{2} \theta+\cos ^{2} \theta+\tan ^{2} \theta $$

Step-by-Step Solution

Answer

The simplification of the trigonometric expression \( \sin^2 \theta + \cos^2 \theta + \tan^2 \theta \) gives the result 1.

Step 1: Recognize the Trigonometric Identities

Firstly, the identity \( \sin^2 \theta + \cos^2 \theta \) is a fundamental trigonometric identity and that is always equal to 1. The identity \( \tan^2 \theta \) can be also simplified using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \).

Step 2: Apply the Trigonometric Identities

Substitute \( \sin^2 \theta + \cos^2 \theta \) with 1 and \( \tan^2 \theta \) with \( \frac{\sin^2 \theta}{\cos^2 \theta} \). This will give: 1 + \( \frac{\sin^2 \theta}{\cos^2 \theta} \).

Step 3: Simplify the Fraction

Since we know that \( \sin^2 \theta + \cos^2 \theta = 1 \), we can replace the numerator of the fraction \( \frac{\sin^2 \theta}{\cos^2 \theta} \) with \( 1 - \cos^2 \theta \). This gives \( 1 + \frac{1 - \cos^2 \theta}{\cos^2 \theta} \). Distributing the \( \cos^2 \theta \) in the denominator of the fraction gives \( 1 + 1 - \rac{\cos^2 \theta}{\cos^2 \theta} \), simplifying to 1 + 1 - 1, which is 1.

Key Concepts

Simplifying Trigonometric Expressions

Understanding how to simplify trigonometric expressions is essential in both algebra and calculus. It involves reducing complex trigonometric expressions to simpler forms that are often easier to understand and work with.
When dealing with expressions like \( \sin^2 \theta + \cos^2 \theta + \tan^2 \theta \), one of the key approaches is finding common trigonometric identities that can help simplify the given expression. These identities act as the rules of the language of trigonometry, allowing for quick transformations between different forms.
To simplify these expressions, it's vital to recognize parts of the formula that can be directly translated into simpler terms using known identities. This makes handling trigonometric problems more manageable and lays a strong foundation for dealing with more complex equations later on.

Fundamental Trigonometric Identities

Fundamental trigonometric identities are a set of equations that are always true for any angle \( \theta \). They are the backbone of simplifying trigonometric expressions.
Some of the most vital identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Tangent Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

A vital part of simplifying expressions is substituting these identities into the expression, transforming complex trigonometric formulas into simpler ones.
For example, if you know \( \sin^2 \theta + \cos^2 \theta = 1 \), you can easily replace the sum of squares in the expression \( \sin^2 \theta + \cos^2 \theta + \tan^2 \theta \) with just "1".
Understanding and memorizing these fundamental identities can immensely ease the process of solving and simplifying trigonometric problems.

Trigonometric Simplification Techniques

Once you're familiar with the trigonometric identities, the next skill to master is the use of simplification techniques. These techniques help in breaking down the algebraic manipulation involved in trigonometric expressions.
Consider the expression \( \sin^2 \theta + \cos^2 \theta + \tan^2 \theta \). To simplify it:

• Recognize \( \sin^2 \theta + \cos^2 \theta \) as 1 using the Pythagorean Identity.
• Substitute \( \tan^2 \theta \) with \( \frac{\sin^2 \theta}{\cos^2 \theta} \) using the tangent identity.
• Further break down the expression using simple algebra to achieve the final simplified form.

By methodically applying these substitutions and simplifications, the expression is reduced to a simpler form. This approach not only simplifies the original expression but also builds a methodical way to approach similar problems, ensuring consistency and accuracy.