Converting Degrees to Radians: Expressing 65°

Understanding how to convert between different units of angle measurement, like degrees and radians, is fundamental in trigonometry and geometry. Degrees are commonly used for measuring angles in everyday situations, while radians are the standard unit in many areas of mathematics and physics, especially calculus.

The relationship between degrees and radians is based on the circumference of a circle. A full circle is 360° or \(2\pi\) radians. This gives us the core conversion factor:

• 180° = \(\pi\) radians

To convert an angle from degrees to radians, we can use the following formula:

Radians = Degrees \(\times \frac{\pi}{180^\circ}\)

Now, let's apply this formula to express 65° in radians.

Step-by-Step Conversion of 65° to Radians

1. Start with the angle given in degrees: 65°.
2. Use the conversion formula: Radians = Degrees \(\times \frac{\pi}{180^\circ}\).
3. Substitute the degree value into the formula: Radians = \(65 \times \frac{\pi}{180}\).
4. Simplify the fraction \(\frac{65}{180}\). Both 65 and 180 are divisible by 5.
5. Divide the numerator by 5: \(65 \div 5 = 13\).
6. Divide the denominator by 5: \(180 \div 5 = 36\).
7. The simplified fraction is \(\frac{13}{36}\).
8. Therefore, 65° expressed in radians is \(\frac{13\pi}{36}\).

So, 65° is equivalent to \(\frac{13\pi}{36}\) radians.

Comparing this result with the given options, we find that \(\frac{13\pi}{36}\) radians matches one of the choices.