Math Problem Statement

Find \( \frac{2 \cdot \tan 30^\circ}{1 + \tan^2 30^\circ} \)

Solution

The expression given in the image is:

2tan301+tan230\frac{2 \cdot \tan 30^\circ}{1 + \tan^2 30^\circ}

Let's break this down and solve it step-by-step.

  1. Step 1: Evaluate tan30\tan 30^\circ

The value of tan30\tan 30^\circ is well-known and equals:

tan30=13\tan 30^\circ = \frac{1}{\sqrt{3}}

  1. Step 2: Substitute into the expression

Now, substitute tan30=13\tan 30^\circ = \frac{1}{\sqrt{3}} into the given equation:

2131+(13)2\frac{2 \cdot \frac{1}{\sqrt{3}}}{1 + \left(\frac{1}{\sqrt{3}}\right)^2}

  1. Step 3: Simplify the denominator

First, simplify the square of 13\frac{1}{\sqrt{3}}:

(13)2=13\left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}

So, the denominator becomes:

1+13=431 + \frac{1}{3} = \frac{4}{3}

  1. Step 4: Simplify the whole expression

Now the expression becomes:

21343=21334\frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{4}{3}} = 2 \cdot \frac{1}{\sqrt{3}} \cdot \frac{3}{4}

Simplify the multiplication:

=643=323= \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}}

  1. Step 5: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and denominator by 3\sqrt{3}:

32333=336=32\frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}

Thus, the simplified result is:

32\boxed{\frac{\sqrt{3}}{2}}

Would you like further details or have any questions on this? Here are some related questions to explore:

  1. How do you compute other trigonometric identities like sin30\sin 30^\circ and cos30\cos 30^\circ?
  2. How do you rationalize a denominator in general?
  3. What are other methods for simplifying trigonometric expressions?
  4. How does the tangent function behave for other common angles like 4545^\circ and 6060^\circ?
  5. Can you solve the same expression for different angles, like 4545^\circ or 6060^\circ?

Tip: Remember, it's useful to memorize key trigonometric values for angles like 3030^\circ, 4545^\circ, and 6060^\circ, as they often appear in problems like this one.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Simplification of Expressions

Formulas

\( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
\( 1 + \tan^2 \theta = \sec^2 \theta \)

Theorems

Tangent Function
Rationalization of Denominators

Suitable Grade Level

Grades 10-12