Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\frac{2 \tan 15^{\circ}}{1-\tan ^{2} 15^{\circ}}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{\sqrt{3}} \).

Step 1: Understanding the Expression

The given expression is \( \frac{2 \tan 15^{\circ}}{1-\tan ^{2} 15^{\circ}} \). We need to simplify this expression using a trigonometric identity related to tangent.

Step 2: Recall Tangent Double Angle Identity

The tangent double angle identity is \( \tan(2 \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \). Identifying this pattern in the given expression, we can see it matches the structure of the double angle identity for \( \theta = 15^{\circ} \).

Step 3: Apply the Identity

Applying the identity \( \tan(2 \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \) to the expression, we recognize that the expression simplifies to \( \tan(2 \times 15^{\circ}) \).

Step 4: Calculate the Double Angle

Since we have identified our expression as \( \tan(30^{\circ}) \), and we know from trigonometric tables that \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), we can write this as the simplified form of the expression.

Step 5: Conclusion of Simplification

The original expression \( \frac{2 \tan 15^{\circ}}{1-\tan ^{2} 15^{\circ}} \) simplifies to \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).

Key Concepts

Tangent Double Angle Identity

The Tangent Double Angle Identity is a useful tool when it comes to simplifying expressions involving tangent. This identity states that \( \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
It's a part of the larger set of trigonometric identities designed to help you express equations more simply. Think of it like a recipe that tells you exactly how to combine your ingredients (in this case, sine and cosine) to get your result.
When you recognize the pattern of the Tangent Double Angle Identity in an equation, it becomes a shortcut to solving complex trigonometric functions. For instance, if you identify your given expression in the form of \( \frac{2 \tan \theta}{1 - \tan^2 \theta} \), simply replace it with \( \tan(2\theta) \).
If the problem states \( \frac{2 \tan 15^{\circ}}{1-\tan ^{2} 15^{\circ}} \),it quickly transforms to \( \tan(30^{\circ}) \), simplifying calculation immediately without cumbersome arithmetic.

Trigonometric Functions

Trigonometric functions are the heart of trigonometry. They relate the angles and sides of a triangle and are fundamental in many areas of geometry and physics. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function provides a unique relation:

• The sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the sine to the cosine of that angle, which is also the ratio of the opposite side to the adjacent side in a right triangle.

These functions extend beyond the simple triangles formed by angles of 0, 30, 45, 60, and 90 degrees, yet these particular angles are key to mastering trigonometry as they offer clean calculation ratios. Understanding these core trigonometric relationships forms a basis for deeper exploration into more advanced areas such as wave functions and Fourier transforms. By using special angles like 15° or 30° along with trigonometric identities, one can simplify and solve complex trigonometric expressions.

Exact Trigonometric Values

Exact trigonometric values are specific values for sine, cosine, and tangent that are well-known and widely used in both geometry and calculus. These values are considered exact because they are derived from geometric principles rather than numerical approximations. Common angles with exact trigonometric values include 0°, 30°, 45°, 60°, and 90°.
For instance, from simple geometric constructions or trigonometric tables, we know:

• \( \sin(30^{\circ}) = \frac{1}{2} \)
• \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)
• \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \)

These numbers remain constant and can be used to simplify expressions without a calculator. In our example, the expression \( \frac{2 \tan 15^{\circ}}{1 - \tan^2 15^{\circ}} \) simplifies to \( \tan(30^{\circ}) \), which we know immediately as \( \frac{1}{\sqrt{3}} \) due to its exact trigonometric value. Mastering these exact values helps students quickly identify and solve problems, making calculations more efficient and precise.