Problem 46 - Algebra and Trigonometry

Question

Use a half-angle formula to find the exact value of each expression. $$ \tan \frac{3 \pi}{8} $$

Step-by-Step Solution

Answer

The exact value of \( \tan(\frac{3\pi}{8}) \) is \( \frac{1}{\sqrt{2}-1} \).

Step 1: Double the Given Angle

The given angle is \( \frac{3\pi}{8} \). Twice this value is \( 2 * \frac{3\pi}{8} = \frac{3\pi}{4} \). This is the \( \theta \) value that will be used.

Step 2: Apply the Cosine Value to Find the Exact Cosine of \( \theta \)

The cosine of angle \( \theta \) is \( \cos(\frac{3\pi}{4}) \). This can be obtained directly from the unit circle or other means. The value is \( -\frac{\sqrt{2}}{2} \).

Step 3: Substitute Cosine Value into Half-angle Formula

Now substitute into the half-angle formula: \( \tan(\frac{3\pi}{8}) = \pm \sqrt{\frac{1 - (-\frac{\sqrt{2}}{2})}{1 + (-\frac{\sqrt{2}}{2})}} = \pm \sqrt{\frac{1+\sqrt{2}/2}{1-\sqrt{2}/2}} \)

Step 4: Simplify and Determine the Correct Sign

By simplifying, we get \( \pm \frac{1}{\sqrt{2}-1} \). The '+/-' comes from considering whether the angle is in a quadrant where tangent is positive or negative. Since \( \frac{3\pi}{8} \) (between \( 0 \) and \( \pi/2 \)), tan is positive, so the answer is \( \frac{1}{\sqrt{2}-1} \).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry, offering a visual representation of angles and their corresponding sine, cosine, and tangent values. By definition, the unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. It helps us understand how trigonometric functions cycle through one full revolution, which corresponds to an angle of \( 2\pi \, \text{radians}\) or \( 360\, \text{degrees}\).

Key points on the unit circle include (1, 0), (0, 1), (-1, 0), and (0, -1). These correspond to angles of 0, \( \frac{\pi}{2}\), \( \pi\), and \( \frac{3\pi}{2}\), respectively.

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

Understanding the unit circle is crucial when solving trigonometric problems because it allows you to quickly find the values of trigonometric functions at key angles. This directly applies to finding exact values using trigonometric identities or formulations such as the half-angle identities.

Tangent Function

The tangent function is one of the fundamental trigonometric functions, defined as the ratio of the sine to the cosine of an angle. Given an angle \( \theta\), the tangent is expressed as:

\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
The tangent function has unique characteristics:

• Its value is undefined when the cosine of the angle is zero (e.g., \( \frac{\pi}{2}\) and \( \frac{3\pi}{2}\)).
• It repeats every \( \pi \, \text{radians}\), making it a periodic function.
• The graph of the tangent function features asymptotes where it is undefined, giving it a distinctive shape.

In solving \( \tan \frac{3\pi}{8} \, \text{using a half-angle formula}\), understanding how the tangent function operates is important. This includes recognizing in which quadrant the angle lies and if the tangent value will be positive or negative. In this case, since \( \frac{3\pi}{8}\) lies in the first quadrant, the tangent value is positive.

Trigonometric Identities

Trigonometric identities are mathematical statements that are true for all values of the variables involved. They are essential tools used to simplify and solve trigonometric equations. In the context of the half-angle formula, these identities provide methods for expressing trigonometric functions of half angles in terms of the original angle.

The half-angle identity for tangent is given by:\[ \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \]
This identity is immensely helpful because:

• It allows us to find the tangent of half an angle if the cosine of the full angle is known.
• The "+/-" indicates that the identity can have a positive or negative value, depending on the angle's quadrant.

Using these identities effectively requires knowledge of other trigonometric functions and their interrelationships within different quadrants. This helps to accurately determine the correct sign and value needed in specific problems.

Exact Values

Exact values refer to solutions of trigonometric functions expressed in terms of integers, fractions, or radicals rather than decimal approximations. When solving for the exact value of trigonometric expressions, it is crucial to adhere strictly to the definitions and identities.

The process involves:

• Using trigonometric identities like the half-angle formulas to express values in a simplified form.
• Employing the unit circle to recognize known angle values and their corresponding sine, cosine, and tangent values.

For example, in the step-by-step solution for finding \( \tan \frac{3\pi}{8}\), the cosine of \( \frac{3\pi}{4}\) was found using the unit circle as \( -\frac{\sqrt{2}}{2}\). This was then substituted into the half-angle formula to derive the exact tangent value, \( \frac{1}{\sqrt{2}-1}\).
Working towards exact values ensures accuracy and precision in mathematical calculations, especially important in fields requiring exact mathematical solutions like engineering and physics.