Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify the expression. $$\frac{2-\tan \theta}{2 \csc \theta-\sec \theta}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\sin \theta\).

Step 1: Simplify the Denominator with Trigonometric Identities

Firstly, let's simplify the denominator: \[2 \csc \theta - \sec \theta.\]Substitute the trigonometric identities:\[\csc \theta = \frac{1}{\sin \theta},\]and\[\sec \theta = \frac{1}{\cos \theta}.\]Thus, the denominator becomes:\[2 \left(\frac{1}{\sin \theta}\right) - \frac{1}{\cos \theta} = \frac{2}{\sin \theta} - \frac{1}{\cos \theta}.\]

Step 2: Find a Common Denominator for the Denominator

To combine the terms in the denominator, we need a common denominator:\[\frac{2}{\sin \theta} - \frac{1}{\cos \theta} = \frac{2\cos \theta - \sin \theta}{\sin \theta \cos \theta}.\]

Step 3: Write the Entire Expression with a Single Denominator

The expression now becomes:\[\frac{2 - \tan \theta}{\frac{2\cos \theta - \sin \theta}{\sin \theta \cos \theta}}.\]Use the reciprocal to simplify further:\[= (2 - \tan \theta) \cdot \frac{\sin \theta \cos \theta}{2\cos \theta - \sin \theta}.\]

Step 4: Simplify the Numerator

Simplify \(2 - \tan \theta\) using the identity:\[\tan \theta = \frac{\sin \theta}{\cos \theta}.\]This gives:\[2 - \tan \theta = 2 - \frac{\sin \theta}{\cos \theta} = \frac{2\cos \theta - \sin \theta}{\cos \theta}.\]

Step 5: Simplify the Entire Expression

Substitute the simplified numerator back:\[\left(\frac{2\cos \theta - \sin \theta}{\cos \theta}\right) \cdot \frac{\sin \theta \cos \theta}{2 \cos \theta - \sin \theta}.\]Cancel \(2\cos \theta - \sin \theta\) from the numerator and denominator:\[\left(\frac{\sin \theta \cos \theta}{\cos \theta}\right).\]\[= \sin \theta.\]

Step 6: Conclusion

The simplified expression is equal to \(\sin \theta\).

Key Concepts

Simplifying Expressions

Working with mathematical expressions often involves making them as simple as possible. This means finding ways to condense the expression without changing its value. By using known identities and algebraic techniques, complicated fractions or combinations can be broken down into more manageable forms. Key strategies for simplifying expressions include:

• Identifying common terms that can be combined.
• Rewriting parts of the expression using mathematical identities.
• Canceling terms that appear in both the numerator and the denominator.
• Rearranging or factoring expressions to reveal simplification opportunities.

By simplifying expressions, not only do you make calculations easier, but you also gain a better understanding of the underlying mathematical relationships. Moreover, a simplified expression is often easier to interpret and can reveal additional insights about the problem at hand.

Trigonometric Simplification

Trigonometric simplification involves using trigonometric identities to reduce trigonometric expressions to simpler forms. This process can often turn a complex trigonometric expression into one that highlights a specific trigonometric function or makes the expression easy to handle in equations and calculations.Consider the exercise at hand, where we simplify \(\frac{2 - \tan \theta}{2 \csc \theta - \sec \theta}\).We start by applying trigonometric identities to simplify its denominator. Using the identities:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)

allows us to transform the denominator from a combination of terms into a single, unified fraction.Afterward, finding a common denominator within that fraction helps us to combine the terms. By strategically applying the trigonometric identities, the expression becomes less cluttered, paving the way to reveal more about its constituent parts. The final outcome of such simplification could be a straightforward result, such as the single-function expression \(\sin \theta\), making further calculations or analysis easier to manage.

Trigonometric Functions

Trigonometric functions are fundamental in trigonometry and include sine, cosine, tangent, cosecant, secant, and cotangent. They describe relationships between the angles and sides of triangles, particularly right-angled triangles. Understanding these functions is crucial for navigating expressions that involve angles and their associated measurements.Key trigonometric functions and their reciprocal identities used in simplifications are:

• Sine \(\sin \theta\) and its reciprocal function cosecant \(\csc \theta = \frac{1}{\sin \theta}\).
• Cosine \(\cos \theta\) and its reciprocal secant \(\sec \theta = \frac{1}{\cos \theta}\).
• Tangent \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and cotangent \(\cot \theta = \frac{1}{\tan \theta}\).

When simplifying expressions like the one in our exercise, these functions and their identities allow for rewriting complex expressions into meaningful, simpler components. Recognizing these relationships is vital, especially since they frequently reappear in various areas of advanced mathematics, physics, and engineering.By understanding the basic definitions and properties of trigonometric functions, learners can unlock powerful techniques for analyzing and solving mathematical problems. This foundational knowledge also enables further exploration into more complex topics involving trigonometry.