Answer :
To solve the problem and find the approximate value of the angle, we are given that the radius intersects the unit circle at the point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex].
1. Understand the Geometry: The point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex] is on the unit circle, which means that it forms a right-angled triangle with the x-axis and the line (radius) from the origin (0, 0) to [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex]. The radius of the unit circle is 1.
2. Identify Triangle Sides:
- The x-coordinate, [tex]\(\frac{3}{5}\)[/tex], represents the length of the base (adjacent side to the angle) of this right triangle.
- The y-coordinate, [tex]\(\frac{4}{5}\)[/tex], represents the length of the height (opposite side to the angle) of the triangle.
- The hypotenuse is the radius, which is 1, as it is the unit circle.
3. Calculate the Angle in Radians:
- We can use the cosine function to find the angle [tex]\(\theta\)[/tex], because [tex]\(\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}\)[/tex].
- Therefore, [tex]\(\theta = \cos^{-1}\left(\frac{3}{5}\right)\)[/tex].
4. Convert to Degrees:
- Once you have the angle in radians, convert it to degrees using the conversion factor: [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex].
5. Approximate the Angle:
- The angle we've calculated is approximately 53.1 degrees in its degree form.
- In radians, it rounds to approximately 0.9 radians.
In conclusion, the approximate value of the angle described by the given point on the unit circle is 53.1 degrees or 0.9 radians.
1. Understand the Geometry: The point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex] is on the unit circle, which means that it forms a right-angled triangle with the x-axis and the line (radius) from the origin (0, 0) to [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex]. The radius of the unit circle is 1.
2. Identify Triangle Sides:
- The x-coordinate, [tex]\(\frac{3}{5}\)[/tex], represents the length of the base (adjacent side to the angle) of this right triangle.
- The y-coordinate, [tex]\(\frac{4}{5}\)[/tex], represents the length of the height (opposite side to the angle) of the triangle.
- The hypotenuse is the radius, which is 1, as it is the unit circle.
3. Calculate the Angle in Radians:
- We can use the cosine function to find the angle [tex]\(\theta\)[/tex], because [tex]\(\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}\)[/tex].
- Therefore, [tex]\(\theta = \cos^{-1}\left(\frac{3}{5}\right)\)[/tex].
4. Convert to Degrees:
- Once you have the angle in radians, convert it to degrees using the conversion factor: [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex].
5. Approximate the Angle:
- The angle we've calculated is approximately 53.1 degrees in its degree form.
- In radians, it rounds to approximately 0.9 radians.
In conclusion, the approximate value of the angle described by the given point on the unit circle is 53.1 degrees or 0.9 radians.