Given the following point on the unit circle, find the angle, to the nearest *tenth of a degree* (if necessary), of the terminal side through that point, 0, degrees, is less than or equal to, theta, is less than, 360, degrees, .0∘≤*θ*<360∘.

P, equals, left parenthesis, start fraction, square root of, 3, end square root, divided by, 5, end fraction, comma, start fraction, square root of, 22, end square root, divided by, 5, end fraction, right parenthesis

*P*=(53​​,522​​)

Given the point \( P = \left( \frac{\sqrt{3}}{5}, \frac{\sqrt{22}}{5} \right) \) on the unit circle, we need to find the angle \( \theta \), where \( 0^\circ \leq \theta < 360^\circ \), that corresponds to this point.

### Step 1: Identify Coordinates
The point \( P \) has coordinates:

\[
x = \frac{\sqrt{3}}{5}, \quad y = \frac{\sqrt{22}}{5}
\]

Since this point is on the unit circle, the angle \( \theta \) can be found by using the inverse trigonometric function of tangent:

\[
\tan(\theta) = \frac{y}{x}
\]

### Step 2: Calculate \( \tan(\theta) \)
Substitute the values of \( x \) and \( y \):

\[
\tan(\theta) = \frac{\frac{\sqrt{22}}{5}}{\frac{\sqrt{3}}{5}} = \frac{\sqrt{22}}{\sqrt{3}} = \sqrt{\frac{22}{3}}
\]

Let's calculate this value numerically.

### Step 3: Find \( \theta \)
Now, we take the arctangent of the result to find the angle in degrees. We also need to ensure that the angle is within the specified range.

I'll calculate this now.Let's proceed with calculating the value manually:

\[
\tan(\theta) = \sqrt{\frac{22}{3}} \approx \sqrt{7.33} \approx 2.71
\]

Now, we apply the arctangent function to this value:

\[
\theta \approx \tan^{-1}(2.71)
\]

Using a calculator, we find:

\[
\theta \approx 69.4^\circ
\]

### Step 4: Determine Quadrant
Both \( x > 0 \) and \( y > 0 \), so the point \( P \) is in the **first quadrant**. Therefore, the angle \( \theta \) is directly \( 69.4^\circ \) since it lies in this quadrant.

Thus, the angle \( \theta \) to the nearest tenth of a degree is:

\[
\boxed{69.4^\circ}
\]

Would you like more details on this process, or do you have any questions?

### Related Questions:
1. How can we convert an angle in degrees to radians?
2. What is the general formula for calculating the tangent of an angle on the unit circle?
3. How do we determine which quadrant a point lies in on the unit circle?
4. What happens if \( \tan(\theta) \) is negative? How does it affect \( \theta \)?
5. How can we use sine or cosine to find angles on the unit circle?

### Tip:
When solving for angles using inverse trigonometric functions, always check which quadrant the point lies in to adjust the angle correctly.

Given the following point on the unit circle, find the angle, to the nearest tenth of a degree, of the terminal side through that point, 0° ≤ θ < 360°. P = (√3/5, √22/5)

Solve for the angle θ in degrees using inverse trigonometry from point P(√3/5, √22/5) on the unit circle. Understand the steps to find the tangent, apply the arctangent function, and determine the quadrant for an accurate solution.

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