\arctan \left(\frac{2}{11}\right)+2\arctan \left(\frac{1}{7}\right)=\arctan \left(\frac{1}{2}\right)

We are asked to verify the equation:

\[
\arctan \left(\frac{2}{11}\right) + 2 \arctan \left(\frac{1}{7}\right) = \arctan \left(\frac{1}{2}\right)
\]

### Step 1: General formula for sum of arctangents
We will use the sum formula for the arctangent function:

\[
\arctan(a) + \arctan(b) = \arctan\left(\frac{a + b}{1 - ab}\right)
\]

as long as \( ab < 1 \).

### Step 2: Simplifying the equation

Let us rewrite the left-hand side \( \arctan \left(\frac{2}{11}\right) + 2 \arctan \left(\frac{1}{7}\right) \). To simplify, we handle \( 2 \arctan \left(\frac{1}{7}\right) \) using the double-angle identity:

\[
2 \arctan(x) = \arctan\left(\frac{2x}{1 - x^2}\right)
\]

For \( x = \frac{1}{7} \), we get:

\[
2 \arctan\left(\frac{1}{7}\right) = \arctan\left(\frac{2 \times \frac{1}{7}}{1 - \left(\frac{1}{7}\right)^2}\right) = \arctan\left(\frac{\frac{2}{7}}{1 - \frac{1}{49}}\right)
\]

This simplifies to:

\[
\arctan\left(\frac{\frac{2}{7}}{\frac{48}{49}}\right) = \arctan\left(\frac{2}{7} \times \frac{49}{48}\right) = \arctan\left(\frac{98}{336}\right) = \arctan\left(\frac{7}{24}\right)
\]

### Step 3: Now we have the equation

\[
\arctan \left(\frac{2}{11}\right) + \arctan \left(\frac{7}{24}\right)
\]

Using the sum formula for arctangents:

\[
\arctan \left(\frac{2}{11}\right) + \arctan \left(\frac{7}{24}\right) = \arctan\left(\frac{\frac{2}{11} + \frac{7}{24}}{1 - \frac{2}{11} \times \frac{7}{24}}\right)
\]

Simplifying the fractions:

\[
\frac{2}{11} + \frac{7}{24} = \frac{48 + 77}{264} = \frac{125}{264}
\]

For the denominator:

\[
1 - \frac{2}{11} \times \frac{7}{24} = 1 - \frac{14}{264} = 1 - \frac{7}{132} = \frac{125}{132}
\]

Thus:

\[
\arctan \left(\frac{\frac{125}{264}}{\frac{125}{132}}\right) = \arctan \left(\frac{125}{264} \times \frac{132}{125}\right) = \arctan\left(\frac{132}{264}\right) = \arctan\left(\frac{1}{2}\right)
\]

### Conclusion:
Both sides of the equation are equal:

\[
\arctan \left(\frac{2}{11}\right) + 2 \arctan \left(\frac{1}{7}\right) = \arctan \left(\frac{1}{2}\right)
\]

The equation is **true**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you derive the sum formula for arctangents?
2. Can we apply the sum formula for values where \(ab \geq 1\)?
3. How does the arctangent double-angle identity work in general?
4. What are the properties of the arctangent function?
5. How can arctangent functions be used in geometry or physics?

**Tip:** The arctangent sum formula is particularly useful in trigonometry when solving equations involving tangent sums or when dealing with angles in non-right triangles.

arctan \left(\frac{2}{11}\right)+2\arctan \left(\frac{1}{7}\right)=arctan \left(\frac{1}{2}\right)

Explore the solution to the equation arctan(2/11) + 2 arctan(1/7) = arctan(1/2) using key trigonometric concepts like the sum formula for arctangents and the double-angle identity. This detailed step-by-step explanation is ideal for students in Grades 11-12 or introductory college courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation