To show that \(\tan 15^\circ = 2\sqrt{3}\), we will use trigonometric identities. Specifically, we'll use the angle subtraction identity for tangent, which states:

\[
\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b}
\]

For this problem, we choose \(a = 45^\circ\) and \(b = 30^\circ\), since \(45^\circ - 30^\circ = 15^\circ\). Let's proceed step by step.

1. **Apply the Identity:**

\[
\tan 15^\circ = \tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \cdot \tan 30^\circ}
\]

2. **Substitute the Values:**

   - \(\tan 45^\circ = 1\)
   - \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

\[
\tan 15^\circ = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}}
\]

3. **Simplify the Expression:**

\[
\tan 15^\circ = \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}}
\]

To simplify this, let's multiply the numerator and the denominator by \(\sqrt{3}\) to eliminate the square roots in the denominator:

\[
\tan 15^\circ = \frac{\sqrt{3} \left(1 - \frac{1}{\sqrt{3}}\right)}{\sqrt{3} \left(1 + \frac{1}{\sqrt{3}}\right)}
\]

\[
\tan 15^\circ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}
\]

4. **Rationalize the Denominator:**

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{3} - 1\):

\[
\tan 15^\circ = \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)}
\]

The denominator simplifies as:

\[
(\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2
\]

The numerator simplifies as:

\[
(\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot 1 + 1^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}
\]

So,

\[
\tan 15^\circ = \frac{4 - 2\sqrt{3}}{2}
\]

\[
\tan 15^\circ = 2 - \sqrt{3}
\]

Now, we see that the calculation shows that \(\tan 15^\circ = 2 - \sqrt{3}\), **not** \(2\sqrt{3}\).

Therefore, the correct value of \(\tan 15^\circ\) is actually \(2 - \sqrt{3}\), not \(2\sqrt{3}\).

Would you like further details or have any questions?

---

Here are some questions related to this topic:

1. What is the angle addition formula for tangent?
2. How would you derive \(\tan 30^\circ\) from a 30-60-90 triangle?
3. Can you prove that \(\tan 45^\circ = 1\) using a unit circle?
4. What are the exact values of sine and cosine for \(15^\circ\)?
5. How do you simplify expressions involving radicals?

**Tip:** Rationalizing the denominator can help simplify trigonometric expressions involving radicals.

Learn how to prove that tan 15 degrees equals 2 root 3 using trigonometric identities and step-by-step calculations. This problem involves trigonometry concepts, specifically the angle subtraction identity for tangent. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation