Find a point on the unit circle whose x-coordinate is 1/5?

To find a point on the unit circle with an \(x\)-coordinate of \( \frac{1}{5} \), we need to recall that the equation of the unit circle is given by:

\[
x^2 + y^2 = 1
\]

Since we know \(x = \frac{1}{5}\), we can substitute this into the equation:

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

Simplifying the equation:

\[
\frac{1}{25} + y^2 = 1
\]

Now, subtract \(\frac{1}{25}\) from both sides:

\[
y^2 = 1 - \frac{1}{25}
\]

\[
y^2 = \frac{25}{25} - \frac{1}{25} = \frac{24}{25}
\]

Finally, take the square root of both sides:

\[
y = \pm \sqrt{\frac{24}{25}} = \pm \frac{\sqrt{24}}{5} = \pm \frac{2\sqrt{6}}{5}
\]

Thus, the two possible points on the unit circle with an \(x\)-coordinate of \(\frac{1}{5}\) are:

\[
\left( \frac{1}{5}, \frac{2\sqrt{6}}{5} \right) \quad \text{and} \quad \left( \frac{1}{5}, -\frac{2\sqrt{6}}{5} \right)
\]

Would you like further clarification on any of the steps, or details about how to apply this solution?

Here are five related questions:
1. How can we find a point on the unit circle if the \(y\)-coordinate is given instead?
2. What are the coordinates of common points on the unit circle like at \(\theta = 0\) and \(\theta = \frac{\pi}{2}\)?
3. How can we use trigonometric functions to describe points on the unit circle?
4. What is the significance of the Pythagorean identity in relation to the unit circle?
5. How does the unit circle relate to trigonometric identities like sine and cosine?

**Tip:** The unit circle is a fundamental tool in trigonometry for understanding the relationship between angles and the coordinates of points on the circle. Knowing how to work with both the geometric and algebraic properties will greatly help in solving trigonometric problems.

Discover how to find points on the unit circle with an x-coordinate of 1/5. This detailed solution involves applying the Pythagorean theorem to solve for the y-coordinate. Ideal for students studying trigonometry in Grades 9-12.

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