Problem 17 - Algebra and Trigonometry

Question

15–26 Use an appropriate half-angle formula to find the exact value of the expression. $$\tan 22.5^{\circ}$$

Step-by-Step Solution

Answer

\(\tan 22.5^{\circ} = \sqrt{2} - 1\).

Step 1: Identify the Half-Angle Formula for Tangent

The half-angle formula for tangent is useful for finding the tangent of an angle that is half of a given angle. The formula is given by: \[\tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \quad \text{or} \quad \frac{\sin \theta}{1 + \cos \theta}\] For this exercise, we are interested in \(\tan 22.5^{\circ}\), which is half of \(45^{\circ}\).

Step 2: Recall the Exact Values of Sine and Cosine for 45°

Recall that \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\). These values will be used in the half-angle formula.

Step 3: Select and Use the Half-Angle Formula

Choose the formula \(\tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta}\) because it typically simplifies neatly with exact values:\[\tan 22.5^{\circ} = \frac{1 - \cos 45^{\circ}}{\sin 45^{\circ}} = \frac{1 - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\]

Step 4: Simplify the Expression

Simplify the fraction:\[\tan 22.5^{\circ} = \frac{1 - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \frac{2 - \sqrt{2}}{\sqrt{2}}\] Multiply both the numerator and the denominator by \(\sqrt{2}\) to rationalize the denominator:\[\tan 22.5^{\circ} = \frac{(2 - \sqrt{2}) \cdot \sqrt{2}}{2}\] Simplifying further gives:\[= \frac{2\sqrt{2} - 2}{2} = \sqrt{2} - 1\]

Step 5: Final Answer

The exact value of \(\tan 22.5^{\circ}\) is \(\sqrt{2} - 1\).

Key Concepts

Tangent

Tangent is one of the primary trigonometric functions, often denoted as \( \tan \). It is particularly useful in right-angle triangle trigonometry and circle-related calculations. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. Therefore:

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

In circular terms, the tangent function represents the slope of the line that intercepts the circle at a single point. This concept is pivotal when dealing with angles that are not part of the typical special angles, like 45°, 30°, or 60°, as it permits us to calculate values for angles such as 22.5° using formulas like the half-angle formula. By understanding tangent in both these contexts, students can gain a deeper comprehension of its application in various mathematical problems.

Exact Value

Calculating the exact value of a tangent or any trigonometric function often involves knowing specific angle values by heart. For angles such as 30°, 45°, and 60°, we can obtain their sine, cosine, and tangent values easily thanks to their symmetry and inherent geometric properties. For instance:

• The exact value of \( \sin 45^{\circ} \) is \( \frac{\sqrt{2}}{2} \).
• The exact value of \( \cos 45^{\circ} \) is \( \frac{\sqrt{2}}{2} \).

These known values are then instrumental when you are working with the half-angle formula. They provide a foundation upon which more complex calculations, like finding \( \tan 22.5^{\circ} \), can be built. By using exact values, a precise solution can be guaranteed, avoiding the approximation errors linked with calculators.

Angle Simplification

Angle simplification is the process of breaking down a given angle into simpler components that are easier to manage mathematically. For instance, when asked to find \( \tan 22.5^{\circ} \), recognizing that this is half of 45° gives you a powerful simplification tool. Through angle simplification:

• You gain insight into which trigonometric identities or formulas are applicable, such as the half-angle formulas.
• You reduce the complexity of solving problems by relying on known values.

Simplifying angles is a strategic approach in trigonometry that ensures more efficient problem solving and a better understanding of the relationships between angles. It enables students to see the bigger picture and link trigonometric problems to patterns and identities they are familiar with.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions—sine, cosine, and tangent—to one another. These identities are vital tools in simplifying trigonometric expressions and solving complex equations. For instance, the half-angle identity for tangent helps calculate the tangent of an angle that is half a known angle. The half-angle identity for tangent can be expressed in two ways:

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

These formulas are derived from other fundamental identities like the Pythagorean identity. By appropriately choosing which form of the half-angle formula to use, calculations can be simplified. Such identities are not only essential for computational purposes but also for a deep understanding of the relationships within trigonometry.