Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\frac{\tan 34^{\circ}}{2\left(1-\tan ^{2} 34^{\circ}\right)}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\tan(22^{\circ})\).

Step 1: Recall Relevant Trigonometric Identities

To simplify the expression, the identity \(1 - \tan^2 \theta = \frac{1}{\sec^2 \theta}\) or the double-angle identity \(\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}\) can be used. We will use the double-angle identity for tangent in this case.

Step 2: Apply the Tangent Double-Angle Identity

The double-angle identity for tangent is \(\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\). Set \(\theta = 34^{\circ}\), then \(2\theta = 68^{\circ}\). Substituting into the identity, we have \(\tan(68^{\circ}) = \frac{2\tan(34^{\circ})}{1-\tan^2(34^{\circ})}\).

Step 3: Simplify the Original Expression

Notice the expression \(\frac{\tan 34^{\circ}}{2(1 - \tan^2 34^{\circ})}\) is a part of the double-angle identity equation. Rearrange to find that \( \frac{\tan 34^{\circ}}{2(1 - \tan^2 34^{\circ})} = \frac{1}{\tan(68^{\circ})} \).

Step 4: Simplify Further Using Angle Properties

Recall that \(\tan(\theta) = \cot(90^{\circ} - \theta)\). Therefore, \(\tan(68^{\circ}) = \cot(22^{\circ}) \), and thus \(\frac{1}{\tan(68^{\circ})} = \tan(22^{\circ})\).

Step 5: Conclude with the Final Answer

The expression simplifies to \(\tan(22^{\circ})\). This is the exact form of the original expression using the identities.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially when dealing with angles and triangles. They provide a way to relate the angles of a triangle to the lengths of its sides. For any angle \(\theta\), there are six primary trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cotangent (\(\cot\)), secant (\(\sec\)), and cosecant (\(\csc\)). These functions are crucial in geometry, engineering, and physics.

Each function has a specific relationship with the sides of a right-angled triangle. For instance:

• Tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
• The cotangent is the reciprocal of the tangent function.
• Secant and cosecant are the reciprocals of cosine and sine respectively.

Understanding these relationships helps in solving problems involving triangles and circular motion.

In addition to defining these functions in triangles, they also have definitions in the context of the unit circle. The unit circle approach helps in extending these functions to all real numbers, allowing angles beyond \(0^{\circ}\) to \(90^{\circ}\).

Double-Angle Identity

The double-angle identity is key in simplifying expressions involving trigonometric functions of double angles. When encountered with an expression involving \(2\theta\), the double-angle identity for tangent comes handy.

The identity \(\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}\) allows us to express \(\tan(2\theta)\) entirely in terms of \(\tan\theta\). This is particularly useful for problems that require expressing a trigonometric function of a doubled angle, such as the original exercise.

Let's use an example: if \(\theta = 34^{\circ}\), applying the double-angle identity means we relate \(\tan(68^{\circ})\) with \(\tan(34^{\circ})\). By substituting \(\theta = 34^{\circ}\) into the identity, you observe how it transforms a more complex expression into a simpler one.

This identity is not unique to tangent; similar identities exist for sine and cosine too, which are helpful for various trigonometric analyses.

Angle Properties

Angle properties are vital tools when working with trigonometric expressions. They allow us to simplify or transform angles to solve equations more efficiently. One of the core properties used in the exercise is the complementary angle property.

According to this property, the tangent of an angle is equal to the cotangent of its complementary angle. Mathematically, this is expressed as \(\tan(\theta) = \cot(90^{\circ} - \theta)\).

This principle was employed in the solution when \(\tan(68^{\circ})\) was rewritten as \(\cot(22^{\circ})\). This not only simplifies expressions but also offers a strategic approach to verify answers or make calculations when direct evaluation seems challenging.

Frequently, understanding these angle properties can shed light on the symmetry and recurring patterns within trigonometric functions, making solving problems less daunting.