Problem 83 - Algebra 2

Question

Simplify each expression. $$ \sin \theta \sec \theta \tan \theta $$

Step-by-Step Solution

Answer

The simplified expression is \( \tan^2 \theta \)

Step 1: Substitute trigonometric identities

Substitute \( \sec \theta \) with its equivalent, which is \( \frac{1}{\cos \theta} \). Also, substitute \( \tan \theta \) with \( \frac{ \sin \theta }{ \cos \theta } \). The original expression now becomes: \( \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{ \sin \theta }{ \cos \theta } \)

Step 2: Simplify by grouping similar terms

Associate similar terms to simplify the expression. The expression then becomes: \( \frac{ \sin^2 \theta }{ \cos^2 \theta } \)

Step 3: Use trigonometric identity again

Recognize that the resulting expression is the same as the square of tangent, \( \tan^2 \theta \). This Final expression is \( \tan^2 \theta \)

Key Concepts

Simplification

Simplifying trigonometric expressions often involves replacing complex terms with equivalent, simpler ones. This allows you to reduce the overall complexity, making the expression easier to understand and work with.

To simplify a trigonometric expression, you should:

• Identify trigonometric identities that can replace current parts of the expression.
• Substitute these new forms into the expression.
• Group and reorganize terms to combine like terms.

After substituting identities, it's essential to review your work. Consider if anything can further be simplified. In some cases, recognizing patterns allows for the application of higher-level identities like Pythagorean or double angle formulas. Ultimately, simplification helps in deriving a form that is easier to analyze or compare.

Trigonometric Functions

Trigonometric functions such as sine (\(\sin \)), cosine (\(\cos \)), and tangent (\(\tan \)) are fundamental in mathematics, especially in relation to angles and the unit circle.

Each function provides valuable information:

• \(\sin \) measures the ratio of the opposite side of an angle to the hypotenuse in a right triangle.
• \(\cos \) measures the ratio of the adjacent side to the hypotenuse.
• \(\tan \) represents the ratio of the opposite side to the adjacent side, which can be directly calculated as \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

These functions extend beyond right triangles. They help model periodic phenomena such as waves, sound, and light, and they are especially useful in engineering and physics. Understanding these functions' identities and how they interrelate allows you to transform and simplify expressions for various applications.

Trigonometric Expressions

Working with trigonometric expressions means dealing with combinations of trigonometric functions. Such expressions can initially appear complex but can often be broken down using known identities.

Here are steps to approach trigonometric expressions:

• Identify any trigonometric identities that match parts of the expression, such as:
  • \(\sec\theta = \frac{1}{\cos\theta}\)
  • \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
• Substitute these into your expression and group similar terms.
• Try reshaping it into something more familiar or simpler like \(\tan^2\theta\).

Successfully simplifying these expressions often involves recognizing familiar patterns and identities, leading to a clearer, more concise result. This process is crucial in calculus and algebra, where simplifying such expressions helps solve larger, more complex equations.