Given: tan A = √7/21,     0° < A < 90°
Find: sin 2A
use 2sinAcosA

We are given \(\tan A = \frac{\sqrt{7}}{21}\) and need to find \(\sin 2A\). The formula for \(\sin 2A\) is:

\[
\sin 2A = 2 \sin A \cos A
\]

### Step 1: Use the identity for \(\tan A\)

\[
\tan A = \frac{\sin A}{\cos A} = \frac{\sqrt{7}}{21}
\]

This can be rewritten as:

\[
\sin A = \frac{\sqrt{7}}{21} \cos A
\]

### Step 2: Use the Pythagorean identity
We know that:

\[
\sin^2 A + \cos^2 A = 1
\]

Substitute \(\sin A = \frac{\sqrt{7}}{21} \cos A\) into this identity:

\[
\left( \frac{\sqrt{7}}{21} \cos A \right)^2 + \cos^2 A = 1
\]

\[
\frac{7}{441} \cos^2 A + \cos^2 A = 1
\]

\[
\cos^2 A \left( \frac{7}{441} + 1 \right) = 1
\]

\[
\cos^2 A \left( \frac{7 + 441}{441} \right) = 1
\]

\[
\cos^2 A \cdot \frac{448}{441} = 1
\]

\[
\cos^2 A = \frac{441}{448}
\]

\[
\cos A = \sqrt{\frac{441}{448}} = \frac{21}{\sqrt{448}} = \frac{21}{8\sqrt{7}}
\]

### Step 3: Find \(\sin A\)

Since \(\sin A = \frac{\sqrt{7}}{21} \cos A\), substitute the value of \(\cos A\):

\[
\sin A = \frac{\sqrt{7}}{21} \cdot \frac{21}{8\sqrt{7}} = \frac{1}{8}
\]

### Step 4: Find \(\sin 2A\)

Now use the formula \(\sin 2A = 2 \sin A \cos A\):

\[
\sin 2A = 2 \cdot \frac{1}{8} \cdot \frac{21}{8\sqrt{7}} = \frac{42}{64\sqrt{7}} = \frac{21}{32\sqrt{7}}
\]

Multiplying both the numerator and denominator by \(\sqrt{7}\) to rationalize the denominator:

\[
\sin 2A = \frac{21\sqrt{7}}{32 \cdot 7} = \frac{3\sqrt{7}}{32}
\]

Thus, the value of \(\sin 2A\) is:

\[
\boxed{\frac{3\sqrt{7}}{32}}
\]

Would you like further details or clarifications?

### Related Questions:
1. How do you find \(\cos 2A\) if \(\tan A\) is known?
2. Can you derive \(\cos A\) directly from \(\tan A\) without using Pythagoras?
3. What is the double angle identity for \(\tan 2A\)?
4. How can you use the unit circle to find \(\sin 2A\)?
5. How would this process change if \(A\) were in a different quadrant?

### Tip:
Always rationalize the denominator when working with square roots to simplify your final answer.

Given: tan A = √7/21, 0° < A < 90°. Find: sin 2A using the formula 2sinAcosA.

Learn how to find sin 2A given tan A = √7/21 using double angle identities and the Pythagorean identity. This solution demonstrates key trigonometric concepts, including step-by-step calculations for sin A and cos A. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation