Problem 38 - Algebra 2

Question

Simplify each trigonometric expression. $$ \frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta} $$

Step-by-Step Solution

Answer

The simplified form of the trigonometric expression \(\frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta}\) is 1.

Step 1: Identify Trigonometric Identities

The first thing to do is to identify each trigonometric identity and their equivalent expressions. So, \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Step 2: Substitute The Trigonometric Identities

The next thing to do is to substitute the trigonometric identities into the original expression. Doing this, we obtain \(\frac{\sin^2 \theta \cdot \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta}}\).

Step 3: Simplify The Expression

Now let's simplify the expression we obtained in step 2 by cancelling out similar terms on the numerator and denominator. This gives us \(\frac{\sin \theta \cdot 1}{\sin \theta} = 1\).

Key Concepts

Simplifying Trigonometric Expressions

When we talk about simplifying trigonometric expressions, we mean making these expressions easier to work with. Often, they involve various trigonometric functions like sine, cosine, tangent, secant, cosecant, and cotangent. In these expressions, terms can sometimes cancel out or combine, making the expression simpler. Simplification helps in solving equations, verifying identities, and evaluating the expressions more efficiently.
To simplify a trigonometric expression like \( \frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta} \), we use known trigonometric identities to rewrite parts of the expression. For example, knowing that \( \csc \theta = \frac{1}{\sin \theta} \), we can replace \( \csc \theta \) in the initial expression, and similarly for \( \sec \theta \) and \( \tan \theta \). This substitution process turns the expression into a form that is easier to simplify because, after substitution, many terms might cancel out or simplify neatly.
The final step involves looking at the new expression and identifying terms that repeat in both the numerator and denominator, allowing them to cancel each other out and often resulting in a much simpler form, sometimes even down to a constant.

Trigonometric Simplification

Trigonometric simplification is the process of reducing trigonometric expressions to their simplest form. This involves using identities like Pythagorean identities, reciprocal identities, and quotient identities. These identities tell us how different trigonometric functions relate to one another and provide ways to transform complex expressions into simpler ones.

• Reciprocal identities include \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).
• Quotient identities such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) are also handy.

In our example, substituting these identities into \( \frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta} \) allows terms to cancel, leading us to a simplified result.
Mastering this skill is useful not just for specific problems but also across many areas in mathematics, physics, and engineering where trigonometry plays an integral part.

Trigonometric Functions

Trigonometric functions are fundamental to understanding relationships in a triangle and modeling periodic phenomena. The basic trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Beyond these, we also have secant (\( \sec \theta \)), cosecant (\( \csc \theta \)), and cotangent (\( \cot \theta \)), which are reciprocals of the primary functions.
These functions are often used in various fields, from calculating angles in architecture and astronomy to analyzing waves in physics and engineering. They form the basis of more complex mathematical and real-world applications.
Recognizing and skilfully using these functions can simplify problems and provide deep insights into the behavior of different physical systems. This is especially true when dealing with trigonometric expressions, where knowing these functions and their identities aids in simplifying and solving them effectively.