Problem 117 - Precalculus

Question

For the following exercises, find the exact value using half-angle formulas. $$ \tan \left(-\frac{3 \pi}{8}\right) $$

Step-by-Step Solution

Answer

\( \tan \left(-\frac{3\pi}{8}\right) = 1 + \sqrt{2} \)

Step 1: Understand the Half-Angle Formula for Tangent

The half-angle formula for tangent is given by \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \) or \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \). You can use either one depending on the angle provided.

Step 2: Identify the Related Angle

We have \( \theta = -\frac{3\pi}{4} \) because \( \tan\left(-\frac{3\pi}{8}\right) = \tan\left(\frac{-3\pi}{4} / 2\right) \). We'll use this larger angle to apply our half-angle identity.

Step 3: Find Cosine and Sine of the Related Angle

Calculate \( \cos\left(-\frac{3\pi}{4}\right) \) and \( \sin\left(-\frac{3\pi}{4}\right) \). Remember that: \( \cos\left(-\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \) and \( \sin\left(-\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \).

Step 4: Apply the Half-Angle Formula

Use the second form of the half-angle formula: \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \). Substitute the values: \( \tan\left(-\frac{3\pi}{8}\right) = \frac{-\frac{1}{\sqrt{2}}}{1 - \frac{1}{\sqrt{2}}} \).

Step 5: Simplify the Expression

Simplify \( \frac{-\frac{1}{\sqrt{2}}}{1 - \frac{1}{\sqrt{2}}} \):\[\tan\left(-\frac{3\pi}{8}\right) = \frac{-\sqrt{2}}{\sqrt{2} - 1}\]Rationalize the denominator to obtain:\[\tan\left(-\frac{3\pi}{8}\right) = \frac{-\sqrt{2}(\sqrt{2} + 1)}{1} = 1 + \sqrt{2}\]

Step 6: Verify the Solution

Check the calculations. The expression simplifies to \(1 + \sqrt{2}\). Since \( \tan \) is periodic with period \( \pi \), the solution is consistent with the function's behavior.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It is often abbreviated as "tan" and is defined as the ratio of the sine function to the cosine function:

• Mathematically, it is expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• This function is periodic with a period of \( \pi \), which means that the tangent of an angle repeats every \( \pi \) radians.
• The tangent function is undefined where the cosine function is zero, leading to vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).

The characteristic shape of the tangent graph is a series of repeating curves with asymptotes disrupting continuity. This behavior makes tangent especially interesting in trigonometric identities and calculations.
It is also important to understand the relationship between tangent and other trigonometric functions when using identities or solving problems involving these."

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any value the variables can take, as long as both sides of the equation are defined.
In the context of half-angle formulas, two key identities are often used:

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

These identities help in finding tangent values of angles that are mathematically half of known angles, simplifying complex calculations.
The use of trigonometric identities is essential for simplifying expressions, proving other identities, and solving trigonometric equations. By utilizing these identities effectively, one can transform and reduce trigonometric expressions to more manageable forms, aiding both in analytical work and practical applications.

Angle Simplification

The process of angle simplification involves manipulating angles to form more manageable or recognizable quantities. In trigonometry, this can happen through the use of half-angle formulas, transformations, and understanding symmetrical properties.

• For example, by expressing an angle as \( \frac{\theta}{2} \), its trigonometric values are computed using half-angle identities.
• This allows us to refer the calculation back to a more familiar angle, such as moving from \( \tan(-\frac{3\pi}{8}) \) to \( \tan(-\frac{3\pi}{4} / 2) \).

By doing angle simplification, one can reduce the complexity of the equation, making calculations simpler and easier to verify. Utilizing symmetry and periodic properties of trigonometric functions can save time and reduce errors. This step is foundational in trigonometry as it allows one to handle angles dynamically and derive exact values necessary for solving precise problems.