Question

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.\(125^{\circ}\)

Short answer

2.18 radians.

Step-by-step solution

Know the Conversion Factor

To convert degrees to radians, use the conversion factor \(\pi \text{ radians} = 180^{\circ}\). This means 1 degree is equal to \(\frac{\pi}{180}\) radians.

Set Up the Conversion Formula

Use the formula \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\). Plug in the given angle in degrees.

Substitute the Given Degree Value

Substitute \(125^{\circ}\) into the conversion formula: \(\text{radians} = 125 \times \frac{\pi}{180}\).

Simplify the Expression

Simplify the expression by multiplying 125 by \(\frac{\pi}{180}\): \(125 \times \frac{\pi}{180} = \frac{125\pi}{180}\).

Calculate the Radians

Convert \(\frac{125\pi}{180}\) to decimal form. Note that \(\pi\) approximately equals 3.14159. Therefore, \(\frac{125 \times 3.14159}{180} = 2.18\).

Key concepts

Degrees to Radians

When we talk about angles, they can be measured in degrees or radians. Degrees and radians are just two different units for measuring angles. Converting from degrees to radians is a common task in trigonometry and calculus. To understand the conversion, it's helpful to remember that a full circle is 360 degrees or 2π radians. This means that 180 degrees is equal to π radians, which leads us to the important concept of the conversion factor.

Conversion Factor

The conversion factor is essential for converting any degree measurement to radians. This factor arises from the relationship between degrees and radians: \(\text{π radians} = 180^\text{°}\). This indicates that 1 degree is equal to \(\frac{π}{180}\text{ radians}\). You can use this conversion factor in a simple formula to convert degrees to radians by multiplying the angle in degrees by \(\frac{π}{180}\). Hence, the formula becomes:
\[\text{radians} = \text{degrees} \times \frac{π}{180}\].

Simplification

The next step is simplifying the expression after substituting the degree value into the formula. For instance, if you want to convert 125° to radians, first input this value into your formula: \[\text{radians} = 125° \times \frac{π}{180}\].
This can be simplified as: \[\text{radians} = \frac{125π}{180}\].
When simplifying, always reduce fractions if possible. This makes the result easier to work with and more understandable.

Decimal Approximation

For practical usage, angles in radians are often presented in decimal form. To convert the radians from fraction form to decimal form, use the approximate value of π (3.14159). Multiply and divide accordingly: \[\frac{125π}{180} ≈ \frac{125 \times 3.14159}{180} \].
When you compute the above expression, you will get approximately 2.18 radians. Always remember to round your final answer to the required decimal places, which in this case are two decimal places.