Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Convert the following degree measures to radians (leave \(\pi\) in your answer). (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(-60^{\circ}\) (d) \(240^{\circ}\) (e) \(-370^{\circ}\) (f) \(10^{\circ}\)

Step-by-Step Solution

Answer

a) \(\frac{\pi}{6}\), b) \(\frac{\pi}{4}\), c) \(-\frac{\pi}{3}\), d) \(\frac{4\pi}{3}\), e) \(-\frac{37\pi}{18}\), f) \(\frac{\pi}{18}\).

Step 1: Understanding the conversion formula

To convert degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula helps convert any angle measured in degrees to radians by multiplying by \( \frac{\pi}{180} \).

Step 2: Convert 30° to Radians

Plug \(30^{\circ}\) into the formula: \( 30 \times \frac{\pi}{180} = \frac{\pi}{6} \). Thus, \(30^{\circ}\) is equal to \(\frac{\pi}{6}\) radians.

Step 3: Convert 45° to Radians

Plug \(45^{\circ}\) into the formula: \( 45 \times \frac{\pi}{180} = \frac{\pi}{4} \). Thus, \(45^{\circ}\) is equal to \(\frac{\pi}{4}\) radians.

Step 4: Convert -60° to Radians

Plug \(-60^{\circ}\) into the formula: \( -60 \times \frac{\pi}{180} = -\frac{\pi}{3} \). Thus, \(-60^{\circ}\) is equal to \(-\frac{\pi}{3}\) radians.

Step 5: Convert 240° to Radians

Plug \(240^{\circ}\) into the formula: \( 240 \times \frac{\pi}{180} = \frac{4\pi}{3} \). Thus, \(240^{\circ}\) is equal to \(\frac{4\pi}{3}\) radians.

Step 6: Convert -370° to Radians

Plug \(-370^{\circ}\) into the formula: \( -370 \times \frac{\pi}{180} = -\frac{37\pi}{18} \). Thus, \(-370^{\circ}\) is equal to \(-\frac{37\pi}{18}\) radians.

Step 7: Convert 10° to Radians

Plug \(10^{\circ}\) into the formula: \( 10 \times \frac{\pi}{180} = \frac{\pi}{18} \). Thus, \(10^{\circ}\) is equal to \(\frac{\pi}{18}\) radians.

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles. It's especially significant in a right-angled triangle, where one of the angles is always 90 degrees. In trigonometry, angles are the key to solving various problems involving triangles and circles.

• Trigonometry involves concepts like sine, cosine, and tangent, which are ratios derived from the sides of a right-angled triangle.
• These ratios are used to define basically how steep a line is, often represented as an angle's value in either degrees or radians.
• Understanding angles is crucial since it affects how these ratios are calculated and interpreted.

In essence, having a grasp of trigonometry helps simplify complex spatial problems, making many advanced math topics more approachable.

Grasping Angle Measurement

When it comes to measuring angles, it can be done in two common units: degrees and radians. Each has its own place in mathematics and everyday life. A degree is a more intuitive measure, commonly used in various daily applications, like weather forecasts and directions. One complete circle is divided into 360 degrees, making each degree a small part of the full rotation. Radians, on the other hand, are mainly used in higher mathematics, like calculus, because they are directly related to the radius of a circle. One full circle in radians is equal to \( 2\pi \) radians. Thus, their utility comes from being a natural measurement in terms of the circle's properties.

• The relationship between degrees and radians is critical, particularly when working with mathematical functions that require angle measurements.
• For conversions, using precise multipliers ensures consistency, such as \( \frac{\pi}{180} \) to convert degrees to radians.

Math Conversion Formulas in Action

Conversion formulas are essential tools in mathematics, particularly when transitioning between different units of measure, like degrees to radians. Knowing how to convert these units is crucial in trigonometric functions where radians are often preferred. To convert a degree angle to radians, we use the formula:\[\text{radians} = \text{degrees} \times \frac{\pi}{180} \]This formula mirrors the relationship between the complete rotations in degrees and radians—where \( 360^{\circ} = 2\pi \) radians. By dividing through by 180, the conversion naturally scales this relationship.

• This process is necessary for solving equations and understanding graphs in trigonometry where angles need to align in compatible units.
• Keeping \( \pi \) in the conversion allows precise answers without approximation, crucial in mathematical proofs or detailed calculations.

Mastering these formula conversions enhances your ability to deal with more complex mathematical tasks and real-world applications.