To find \(\sin(2a)\) given \(\sin(a) = \frac{\sqrt{5}}{5}\), we can use the double-angle identity for sine:

\[
\sin(2a) = 2\sin(a)\cos(a)
\]

First, we need to find \(\cos(a)\). Since we know \(\sin(a)\), we can use the Pythagorean identity:

\[
\sin^2(a) + \cos^2(a) = 1
\]

Substituting the value of \(\sin(a)\):

\[
\left(\frac{\sqrt{5}}{5}\right)^2 + \cos^2(a) = 1
\]

\[
\frac{5}{25} + \cos^2(a) = 1
\]

\[
\frac{1}{5} + \cos^2(a) = 1
\]

\[
\cos^2(a) = 1 - \frac{1}{5} = \frac{4}{5}
\]

Now, taking the square root to find \(\cos(a)\):

\[
\cos(a) = \pm\frac{\sqrt{4}}{\sqrt{5}} = \pm\frac{2}{\sqrt{5}}
\]

The sign of \(\cos(a)\) depends on the quadrant where angle \(a\) is located. However, since we are not given specific information about the quadrant, we'll keep both possibilities for now.

Finally, substitute \(\sin(a)\) and \(\cos(a)\) into the double-angle identity:

\[
\sin(2a) = 2 \times \frac{\sqrt{5}}{5} \times \pm\frac{2}{\sqrt{5}} = \pm\frac{4}{5}
\]

So, \(\sin(2a) = \pm\frac{4}{5}\).

Would you like more details or have any other questions?

### Related Questions:
1. How can we determine the exact sign of \(\cos(a)\)?
2. What is the value of \(\cos(2a)\) given \(\sin(a) = \frac{\sqrt{5}}{5}\)?
3. How do we find \(\tan(2a)\) given \(\sin(a)\)?
4. How do we express \(\sin(4a)\) in terms of \(\sin(a)\)?
5. What is the general formula for \(\sin(na)\) for any integer \(n\)?

**Tip:** When solving trigonometric identities, always consider the quadrant to determine the correct sign of your result.

Learn how to find sin 2a given sin a = √5/5 using trigonometric identities and Pythagorean theorem. Understand the steps involved in solving for sin 2a with detailed explanations. Suitable for high school students studying trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation