If angle theta lies in Quadrant II and sin theta = 3/√45. Determine possible coordinates for point P on the terminal arm of angle theta

Given that angle \(\theta\) lies in Quadrant II and \(\sin \theta = \frac{3}{\sqrt{45}}\), we need to find the possible coordinates of point \(P(x, y)\) on the terminal arm of the angle \(\theta\).

### Step 1: Simplify \(\sin \theta\)
First, simplify the given sine value:
\[
\sin \theta = \frac{3}{\sqrt{45}} = \frac{3}{\sqrt{9 \times 5}} = \frac{3}{3\sqrt{5}} = \frac{1}{\sqrt{5}}
\]
So, \(\sin \theta = \frac{1}{\sqrt{5}}\).

### Step 2: Use the identity \(\sin \theta = \frac{y}{r}\)
Since \(\sin \theta = \frac{y}{r}\), where \(r\) is the radius (distance from the origin) and \(y\) is the y-coordinate:
\[
\frac{y}{r} = \frac{1}{\sqrt{5}}
\]
This implies:
\[
y = \frac{r}{\sqrt{5}}
\]

### Step 3: Find the value of \(r\)
Using the Pythagorean identity \(r^2 = x^2 + y^2\), substitute \(y = \frac{r}{\sqrt{5}}\):
\[
r^2 = x^2 + \left(\frac{r}{\sqrt{5}}\right)^2
\]
\[
r^2 = x^2 + \frac{r^2}{5}
\]
Multiply both sides by 5 to clear the fraction:
\[
5r^2 = 5x^2 + r^2
\]
\[
4r^2 = 5x^2
\]
\[
r^2 = \frac{5x^2}{4}
\]

### Step 4: Substitute \(r\) back into \(y = \frac{r}{\sqrt{5}}\)
Now, substitute \(r^2 = \frac{5x^2}{4}\) into \(y = \frac{r}{\sqrt{5}}\):
\[
y = \frac{\sqrt{\frac{5x^2}{4}}}{\sqrt{5}} = \frac{\sqrt{5x^2}}{2\sqrt{5}} = \frac{x}{2}
\]

### Step 5: Determine the sign of \(x\) and \(y\)
In Quadrant II:
- \(x\) is negative.
- \(y\) is positive.

So, the coordinates of point \(P(x, y)\) are:
\[
P\left(-\frac{2\sqrt{5}}{5}, \frac{\sqrt{5}}{2}\right)
\]

### Final Answer:
Possible coordinates for the point \(P(x, y)\) on the terminal arm of angle \(\theta\) are \(P\left(-\frac{2\sqrt{5}}{5}, \frac{\sqrt{5}}{2}\right)\).

Would you like any further details or clarifications on this?

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Here are 5 related questions you might find interesting:
1. How does the cosine of angle \(\theta\) change in different quadrants?
2. What is the value of \(\cos \theta\) for this particular point \(P\)?
3. How would the coordinates change if \(\theta\) were in Quadrant III?
4. What are the exact coordinates of \(P\) if \(\sin \theta = \frac{2}{\sqrt{13}}\) in Quadrant II?
5. Can you find the tangent of \(\theta\) for the given coordinates of \(P\)?

**Tip:** Always remember that in Quadrant II, the sine function is positive, while the cosine function is negative.

Explore how to determine the coordinates of point P on the terminal arm of angle theta in Quadrant II using trigonometric identities and the Pythagorean identity. Understand how sin theta relates to the coordinates (x, y) and find the exact coordinates based on given trigonometric values.

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