Problem 41 - Precalculus: Functions and Graphs

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: Express Trigonometric Functions in Terms of Sine and Cosine

First, convert all trigonometric functions in the expression \(\frac{2 - \tan \theta}{2 \csc \theta - \sec \theta}\) into sine and cosine. Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\sec \theta = \frac{1}{\cos \theta}\). Substituting these into the expression, we get: \[\frac{2 - \frac{\sin \theta}{\cos \theta}}{2\left(\frac{1}{\sin \theta}\right) - \frac{1}{\cos \theta}} = \frac{2 - \frac{\sin \theta}{\cos \theta}}{\frac{2}{\sin \theta} - \frac{1}{\cos \theta}}.\]

Step 2: Simplify the Numerator

Simplify the numerator \(2 - \frac{\sin \theta}{\cos \theta}\) by getting a common denominator. This becomes \(\frac{2\cos \theta - \sin \theta}{\cos \theta}\).

Step 3: Simplify the Denominator

Similarly, simplify the denominator \(\frac{2}{\sin \theta} - \frac{1}{\cos \theta}\) by getting a common denominator. The denominator becomes \(\frac{2\cos \theta - \sin \theta}{\sin \theta \cos \theta}\).

Step 4: Divide the Fractions

The expression can now be simplified by dividing the simplified numerator by the simplified denominator, which involves inverting the second fraction and multiplying. This gives: \[\frac{2\cos \theta - \sin \theta}{\cos \theta} \times \frac{\sin \theta \cos \theta}{2\cos \theta - \sin \theta} = \frac{\sin \theta \cos \theta}{\cos \theta}. \]

Step 5: Cancel Out Common Terms

Notice that the terms \(2\cos \theta - \sin \theta\) in the numerator and denominator will cancel out, leaving: \[\frac{\sin \theta \cos \theta}{\cos \theta} = \sin \theta.\]

Step 6: Final Simplified Expression

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

Key Concepts

Simplifying Trigonometric Expressions

Trigonometric identities are an essential tool in simplifying expressions involving functions like sine, cosine, and tangent. Simplifying expressions means rewriting them in a more straightforward or more familiar form, often for easier computation or to solve equations. Here, we took an initial complex expression \( \frac{2-\tan \theta}{2 \csc \theta-\sec \theta} \), which involves several trigonometric functions, and reduced it to something elementary.

To start, we expressed the tangential, secant, and cosecant functions in terms of sine and cosine. Simplification might involve finding common denominators in the fractions or canceling out terms when possible. Writing everything using sine and cosine creates common ground for further simplification, since these are the most basic trigonometric functions. This approach underscores the simplicity behind the complexity, revealing that sometimes, what's complex can be reduced back to fundamentals.

Sine and Cosine Functions

Sine and cosine are the foundational trigonometric functions that form the backbone of many mathematical processes, especially in trigonometric identities and equations.

Knowing that:

• \( \sin \theta \) is the opposite side over the hypotenuse in a right triangle.
• \( \cos \theta \) is the adjacent side over the hypotenuse.
• They are periodic with a usual period of \(2\pi\).

Expressing other trigonometric functions in terms of sine and cosine is a strategic step toward simplification, often making it easier to identify like terms and cancel them in equations. For instance, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \csc \theta = \frac{1}{\sin \theta} \).

Using these conversions in our expression allowed us better control over the simplification process, guiding us directly to the answer. This illustrates how often in mathematics, it is beneficial to "speak the same language" – and for trigonometry, that language is often sine and cosine.

Common Denominator in Fraction Simplification

Finding a common denominator is a crucial step when simplifying expressions with fractions, especially when dealing with trigonometric terms. This makes it possible to combine fractions in a meaningful way, allowing you to simplify more effectively.

In our example, once all functions were expressed in terms of sine and cosine, we tackled both the numerator \(2 - \frac{\sin \theta}{\cos \theta}\) and the denominator \( \frac{2}{\sin \theta} - \frac{1}{\cos \theta} \) by identifying common denominators.

The numerator became \( \frac{2\cos \theta - \sin \theta}{\cos \theta} \) and the denominator \( \frac{2\cos \theta - \sin \theta}{\sin \theta \cos \theta} \). Achieving a common denominator allowed the subtraction to proceed smoothly, leading to the expression being reduced to a much simpler form.

Afterwards, dividing the fractions is performed by multiplying the first fraction by the reciprocal of the second. This effectively cancels out terms to yield the simple result: \( \sin \theta \). Creating common ground within fractions streamlines operations, illustrating how structured methods replace initial complexity with clarity.