Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\tan \left(-\frac{\pi}{8}\right)$$

Step-by-Step Solution

Answer

\( \tan \left(-\frac{\pi}{8}\right) = \sqrt{2} - 1 \).

Step 1: Identify the Half-Angle Identity for Tangent

The half-angle identity for tangent is given by \( \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \) or \( \tan \left( \frac{\theta}{2} \right) = \frac{\sin \theta}{1 + \cos \theta} \). These identities will be used to find the expression for \( \tan \left(-\frac{\pi}{8}\right) \).

Step 2: Determine the Angle \( \theta \) for Half-Angle Identity

Given \( \frac{\theta}{2} = -\frac{\pi}{8} \), we need to find \( \theta \). Solving \( \frac{\theta}{2} = -\frac{\pi}{8} \) gives \( \theta = -\frac{\pi}{4} \).

Step 3: Calculate \( \cos\theta \) and \( \sin\theta \)

For \( \theta = -\frac{\pi}{4} \), we have \( \cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \) because cosine is an even function and sine is an odd function.

Step 4: Apply Half-Angle Identity to Find \( \tan \left(-\frac{\pi}{8}\right) \)

Using \( \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \), substitute \( \cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \). Therefore, \( \tan \left(-\frac{\pi}{8}\right) = \frac{1 - \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} \).

Step 5: Simplify the Expression

Simplify the expression: \( \frac{1 - \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} \). First, simplify the numerator: \( 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \). So the expression becomes \( \frac{\frac{2 - \sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = \frac{2 - \sqrt{2}}{-\sqrt{2}} \).

Step 6: Final Simplification Step

Multiply the numerator and denominator by \( \sqrt{2} \) to rationalize: \( \frac{2 - \sqrt{2}}{-\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{(2\sqrt{2} - 2)}{(-2)} \). Therefore, \( \tan \left(-\frac{\pi}{8}\right) = \sqrt{2} - 1 \).

Key Concepts

Half-Angle Identity

Understanding half-angle identities is crucial in trigonometry as they allow us to find the trigonometric functions of half-angles using known values. The half-angle identity for the tangent function can be expressed in two forms:

• \( \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \)
• \( \tan \left( \frac{\theta}{2} \right) = \frac{\sin \theta}{1 + \cos \theta} \)

These identities derive from using known cosine and sine values of the full angle \( \theta \). For example, if you're computing \( \tan \left(-\frac{\pi}{8}\right) \), you see it's a half-angle. Solving \( \frac{\theta}{2} = -\frac{\pi}{8} \) gives \( \theta = -\frac{\pi}{4} \).

By applying the half-angle identity, we resolve the expression using already known values at this particular angle. This method is widely used since it simplifies complex angles to easier and already computed values.

Thus, enabling us to accurately determine trigonometric functions of lesser-used or complex angles without resorting to calculators.

Tangent Function

The tangent function, often denoted as \( \tan \), is one of the primary trigonometric functions. It relates an angle of a right-angled triangle to the ratios of two sides of the triangle. Specifically:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Furthermore, in terms of sine and cosine, the tangent function is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

This definition is particularly useful when using identities like the half-angle identity. It's important to remember that \( \tan \theta \) can be undefined when \( \cos \theta = 0 \) since division by zero is not possible.

In our exercise scenario, we computed \( \tan \left(-\frac{\pi}{8}\right) \). Here the angle is negative, reflecting tangent's periodic and odd nature, meaning \( \tan(-\theta) = -\tan(\theta) \).

The tangent function is crucial for understanding more complex trigonometric relationships and problem-solving scenarios.

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry and include sine, cosine, and tangent. These functions help us relate angles to side lengths in right-angled triangles, and they also extend beyond these into wave modeling and complex number studies.

Each function has an association with the circle's coordinate plane:

• Sine \( \sin \theta \): Represents the y-coordinate of a point on the unit circle.
• Cosine \( \cos \theta \): Represents the x-coordinate, showing the adjacent side compared to the hypotenuse.
• Tangent \( \tan \theta \): The ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

These relationships are not just limited to static right-angled triangle scenarios and are vital for understanding periodic functions, waves, harmonic motion, and even new fields such as complex numbers.

In the exercise, trigonometric functions enabled us to determine \( \tan \left(-\frac{\pi}{8}\right) \) more easily through identities and known values.

Through understanding trigonometric functions deeply, we become adept at solving diverse mathematical problems and gain insights into the behavior of angles and their respective triangles.