Problem 22 - Algebra and Trigonometry

Question

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$ \frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}} $$

Step-by-Step Solution

Answer

The expression is equal to the \( \tan(\frac{\pi}{4}) \) which has an exact value of 1.

Step 1: Identify the Double Angle Identity

We will focus on the tan double angle identity which is known to be \( \tan(2A)=\frac{2\tan(A)}{1-\tan^2(A)} \). We can see that the provided formula corresponds to this identity.

Step 2: Use the Identity to Rewrite the Expression

We place the provided angle \( \frac{\pi}{8} \) into our identity. This gives us \( \tan(2(\frac{\pi}{8})) \), which further simplifies to \( \tan(\frac{\pi}{4}) \).

Step 3: Calculate the Exact Value

The exact value of \( \tan(\frac{\pi}{4}) \) is known to be 1.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that help in simplifying complex expressions and solving equations. They are equations that involve the trigonometric functions sine, cosine, tangent, and others. These identities hold true for every value of the occurring variables, making them powerful in mathematical problem-solving.

One well-known group of trigonometric identities includes the double-angle identities. These are particularly useful because they relate the trigonometric function of double an angle to the function of the single angle. For instance, the tangent double-angle identity is given by:

• \( \tan(2A) = \frac{2\tan(A)}{1-\tan^2(A)} \)

This identity allows us to express complex trigonometric calculations in simpler forms, effectively reducing computation steps in problems related to angles. Understanding and memorizing such identities enable students to quickly identify patterns and make quick analytical decisions in both theoretical and applied mathematical contexts.

Tangent Function

The tangent function, often written as \( \tan \), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function:

• \( \tan(A) = \frac{\sin(A)}{\cos(A)} \)

The tangent function exhibits unique properties such as periodicity and it is undefined for angles where the cosine is zero, as this would lead to division by zero.

It is particularly noteworthy because of its connection to the angle of inclination in right-angled triangles, commonly representing the slope or gradient. In trigonometric identities, the tangent function often helps simplify the process of understanding angle relations, especially through identities like the double angle or sum-to-product identities. 

The tangent function is critical in many real-world applications, including physics and engineering, where it can describe wave behaviors, projectile motion, and even in design fields where angles and slopes are fundamental.

Exact Values in Trigonometry

In trigonometry, certain angle measures have exact trigonometric values which facilitate precise calculations. These are often derived through geometric methods or by utilizing the unit circle. For example, angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) have well-known trigonometric values for sine, cosine, and tangent.

Understanding these exact values is essential as they allow for quick reference and accuracy in solving trigonometric problems without resorting to numerical approximations. For instance, \( \tan(\frac{\pi}{4}) = 1 \), owing to the symmetrical properties of the unit circle and the properties of equivalent triangle sides.

Recognizing these values makes it easier to solve trigonometric equations and verify identities, providing a cornerstone in preparing for higher-level mathematics where these skills are regularly required. It also extends into practical fields, such as engineering and computer science, where the knowledge of these exact values can optimize calculations and algorithms.