Question

Convert each degree measure to radians. Express your answers as exact values and as approximate measures, to the nearest hundredth of a radian. a) \(60^{\circ}\) b) \(150^{\circ}\) c) \(-270^{\circ}\) d) \(72^{\circ}\) e) \(-14.8^{\circ}\) f) \(540^{\circ}\)

Short answer

60° = π/3 radians (1.05 radians). 150° = 5π/6 radians (2.62 radians). -270° = -3π/2 radians (-4.71 radians). 72° = 2π/5 radians (1.26 radians). -14.8° = -37π/450 radians (-0.26 radians). 540° = 3π radians (9.42 radians).

Step-by-step solution

Understand the Conversion Formula

To convert degrees to radians, use the formula: \[\text{Radians} = \text{Degrees} \times \frac{\text{π}}{180}\]

Convert 60 Degrees to Radians

Using the formula, \[60^\text{°} \times \frac{\text{π}}{180} = \frac{\text{π}}{3}\] The approximate value is \[\frac{\text{π}}{3} \approx 1.05 \text{ radians}\]

Convert 150 Degrees to Radians

Using the formula, \[150^\text{°} \times \frac{\text{π}}{180} = \frac{5\text{π}}{6}\] The approximate value is \[\frac{5\text{π}}{6} \approx 2.62 \text{ radians}\]

Convert -270 Degrees to Radians

Using the formula, \[-270^\text{°} \times \frac{\text{π}}{180} = -\frac{3\text{π}}{2}\] The approximate value is \[-\frac{3\text{π}}{2} \approx -4.71 \text{ radians}\]

Convert 72 Degrees to Radians

Using the formula, \[72^\text{°} \times \frac{\text{π}}{180} = \frac{2\text{π}}{5}\] The approximate value is \[\frac{2\text{π}}{5} \approx 1.26 \text{ radians}\]

Convert -14.8 Degrees to Radians

Using the formula, \[-14.8^\text{°} \times \frac{\text{π}}{180} = -\frac{37\text{π}}{450}\] The approximate value is \[-\frac{37\text{π}}{450} \approx -0.26 \text{ radians}\]

Convert 540 Degrees to Radians

Using the formula, \[540^\text{°} \times \frac{\text{π}}{180} = 3\text{π}\] The approximate value is \[3\text{π} \approx 9.42 \text{ radians}\]

Key concepts

Angle Conversion

Angles can be measured in degrees or radians. Converting from one unit to another helps us understand problems in mathematics, physics, and engineering. Degrees are a common measure, typically used in everyday contexts like angles in geometry and geography. Radians, however, are often used in higher-level mathematics, including calculus and trigonometry. To convert between these units, we use specific formulas that relate degrees to radians and vice versa.

Degrees to Radians Formula

To convert an angle from degrees to radians, we use the formula \(\text{Radians} = \text{Degrees} \times \frac{\text{π}}{180}\). This works because 180 degrees is equal to π radians. For example, to convert 60 degrees to radians, we calculate: \(\text{Radians} = 60 \times \frac{\text{π}}{180} = \frac{\text{π}}{3}\). Approximate this using the value of π (approximately 3.14), we get \(\frac{\text{π}}{3} \) which is about 1.05 radians. This formula is essential for translating problems from degree measures (which are often more intuitive) to radian measures (which are more useful in advanced math).

Trigonometry Basics

Trigonometry is the study of relationships between the angles and sides of triangles. In many trigonometry problems, angles are given in degrees, but calculations are often easier or required to be done in radians. Common functions in trigonometry include sine, cosine, and tangent, which all have different values depending on whether the angle is measured in degrees or radians. Understanding how to convert between these units is crucial for solving trigonometric equations and modeling periodic phenomena in engineering and science.

Exact Values in Radians

Sometimes, it’s important to express angle measures as exact values in radians. This is especially true in problems where precision is necessary. For instance, \(60^{\text{°}} = \frac{\text{π}}{3} \) is an exact value. These exact values help us communicate specific angles without rounding errors. For instance, common angles like \(\frac{\text{π}}{6}\), \(\frac{\text{π}}{4}\), and \(\frac{\text{π}}{3}\) occur frequently in trigonometry and are usually remembered as standard values.

Approximate Values in Radians

When working with radians, we sometimes need approximate values. This is useful when precision to a few decimal places is sufficient. For example, \(\frac{\text{π}}{3} \) is exactly 1.047197551, but often rounded to 1.05 radians. These approximations make quick calculations easier. In practical applications like physics, engineers might use these rounded values to simplify complex formulas without significantly impacting accuracy.