Answer :
To find the approximate value of [tex]\(\theta\)[/tex], let's consider the point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex] which lies on the unit circle. This point corresponds to a specific angle [tex]\(\theta\)[/tex] in the unit circle.
1. Understand the Point and its Relation to the Unit Circle:
- In the unit circle, a point [tex]\((x, y)\)[/tex] on the circle can be described using trigonometric functions as [tex]\((\cos(\theta), \sin(\theta))\)[/tex].
- For the point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex], [tex]\(\cos(\theta) = \frac{3}{5}\)[/tex] and [tex]\(\sin(\theta) = \frac{4}{5}\)[/tex].
2. Calculate [tex]\(\theta\)[/tex] in Radians:
- To find [tex]\(\theta\)[/tex] in radians, we use the inverse trigonometric function [tex]\(\cos^{-1}(x)\)[/tex], where [tex]\(x = \frac{3}{5}\)[/tex].
- This will give us [tex]\(\theta \approx 0.9273\)[/tex] radians.
3. Convert [tex]\(\theta\)[/tex] from Radians to Degrees:
- To convert an angle from radians to degrees, use the conversion factor [tex]\(180^\circ/\pi\)[/tex].
- Applying this gives [tex]\(\theta \approx 53.13\)[/tex] degrees.
4. Choose the Closest Approximate Value:
- Based on the options provided, [tex]\(53.1\)[/tex] degrees is the closest to our calculated value of [tex]\(\theta = 53.13\)[/tex] degrees.
Therefore, the approximate value of [tex]\(\theta\)[/tex] is [tex]\(53.1\)[/tex] degrees.
1. Understand the Point and its Relation to the Unit Circle:
- In the unit circle, a point [tex]\((x, y)\)[/tex] on the circle can be described using trigonometric functions as [tex]\((\cos(\theta), \sin(\theta))\)[/tex].
- For the point [tex]\(\left(\frac{3}{5}, \frac{4}{5}\right)\)[/tex], [tex]\(\cos(\theta) = \frac{3}{5}\)[/tex] and [tex]\(\sin(\theta) = \frac{4}{5}\)[/tex].
2. Calculate [tex]\(\theta\)[/tex] in Radians:
- To find [tex]\(\theta\)[/tex] in radians, we use the inverse trigonometric function [tex]\(\cos^{-1}(x)\)[/tex], where [tex]\(x = \frac{3}{5}\)[/tex].
- This will give us [tex]\(\theta \approx 0.9273\)[/tex] radians.
3. Convert [tex]\(\theta\)[/tex] from Radians to Degrees:
- To convert an angle from radians to degrees, use the conversion factor [tex]\(180^\circ/\pi\)[/tex].
- Applying this gives [tex]\(\theta \approx 53.13\)[/tex] degrees.
4. Choose the Closest Approximate Value:
- Based on the options provided, [tex]\(53.1\)[/tex] degrees is the closest to our calculated value of [tex]\(\theta = 53.13\)[/tex] degrees.
Therefore, the approximate value of [tex]\(\theta\)[/tex] is [tex]\(53.1\)[/tex] degrees.
