Question

The following angles are given in degrees. Convert them to radians: \(20^{\circ}, 35^{\circ}, 80^{\circ}, 270^{\circ}\).

Short answer

\( \frac{\pi}{9}, \frac{7\pi}{36}, \frac{4\pi}{9}, \frac{3\pi}{2} \) radians.

Step-by-step solution

Understanding Degree to Radian Conversion

To convert degrees to radians, use the formula: \( radians = degrees \times \frac{\pi}{180} \). This formula transforms an angle measured in degrees to its equivalent in radians by multiplying by the ratio of \( \pi \) to 180.

Convert 20 Degrees to Radians

Using the conversion formula: \( 20^{\circ} \times \frac{\pi}{180} = \frac{20\pi}{180} = \frac{\pi}{9} \) radians. This is the angle 20 degrees expressed in radians.

Convert 35 Degrees to Radians

Using the conversion formula: \( 35^{\circ} \times \frac{\pi}{180} = \frac{35\pi}{180} = \frac{7\pi}{36} \) radians. This is the angle 35 degrees expressed in radians.

Convert 80 Degrees to Radians

Using the conversion formula: \( 80^{\circ} \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \) radians. This is the angle 80 degrees expressed in radians.

Convert 270 Degrees to Radians

Using the conversion formula: \( 270^{\circ} \times \frac{\pi}{180} = \frac{270\pi}{180} = \frac{3\pi}{2} \) radians. This is the angle 270 degrees expressed in radians.

Key concepts

Angle Measurement

Angles are fundamental elements in geometry and trigonometry, and they are used to describe the rotation around a central point. These angles are often measured in two common units: degrees and radians.

Degrees, denoted by the symbol \( ^{\circ} \), are a familiar unit because they divide a circle into 360 equal parts. Each degree represents a tiny fraction of a complete circle. For instance, a right angle is 90 degrees, which is a quarter of a circle.

Radians, on the other hand, offer a more natural measure for angles used in calculus and trigonometry. In radians, angles are expressed as the length of the arc that they "cut" from the unit circle. One full circle is equal to \( 2\pi \) radians. This means \( \pi \) radians make up a half-circle or 180 degrees.

• A complete circle = 360 degrees = \( 2\pi \) radians.
• Half a circle = 180 degrees = \( \pi \) radians.

Radian Conversion Formula

The radian conversion formula is crucial for switching from the degree measurement system to the radian system. The relationship between the two systems is established through the formula:

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

This formula is derived from the fact that \( 180 \) degrees is equivalent to \( \pi \) radians. To convert any angle from degrees to radians, simply multiply the amount in degrees by \( \frac{\pi}{180} \).

For example, to convert \( 20^{\circ} \) to radians, the calculation is:
\[ 20^{\circ} \times \frac{\pi}{180} = \frac{20\pi}{180} = \frac{\pi}{9} \]This step-by-step conversion ensures that you can accurately convert any degree measurement into radians, a vital skill in further mathematics and sciences.

Trigonometry in Physics

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. Its principles are extensively used in physics to understand various phenomena and solve problems efficiently.

Physics often involves the study of waves, oscillations, and periodic phenomena, all of which make extensive use of angles measured in radians. The radian measure, with its natural relation to the arc and circle, allows for more straightforward calculations, especially when dealing with trigonometric functions like sine and cosine.

• Radians simplify the mathematical expressions used in angular velocity calculations.
• They help express angular displacements and accelerations more naturally.
• Wave equations and their properties, such as frequency and phase shift, are efficiently conveyed in radians.

In oscillations or circular motion, understanding how to measure these angles in radians can provide more precise insights and predictions, which is why mastering this conversion is essential in physics.