Problem 62 - Precalculus : Building Concepts and Connections

Question

Convert each angle from degrees to radians. $$-405^{\circ}$$

Step-by-Step Solution

Answer

The equivalent of -405 degrees in radians is \(-\frac{9\pi}{4}\).

Step 1: Understanding the Conversion Ratio

The conversion ratio from degrees to radians is given by \(\frac{\pi}{180}\). In other words, one degree is equivalent to \(\frac{\pi}{180}\) radians.

Step 2: Applying the Conversion Ratio

In order to convert -405 degrees into radians, the given degrees will be multiplied by \(\frac{\pi}{180}\). Therefore, it would be \(-405 \times \frac{\pi}{180}\).

Step 3: Simplifying the Expression

Simplify the given expression to obtain the resulting radians. This would give \(-\frac{9\pi}{4}\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores relationships involving angles and lengths of triangles. It is widely used in various fields such as engineering, physics, and computer science. Trigonometry is founded on the ratios between different sides of a right-angled triangle: sine, cosine, and tangent.
These trigonometric functions are essential in the study of waves, oscillations, and other phenomena in science. When dealing with angles, it is crucial to understand how angles can be measured, which leads us to the concept of angle measurement.

Angle Measurement

Angles can be measured in several units, the most common being degrees and radians. Understanding these units is vital for solving geometry and trigonometry problems.
Here’s a quick overview:

• Degrees: A circle is divided into 360 degrees. This is the most common unit used in everyday life.
• Radians: A radian is based on the radius of the circle. One radian is the angle created when the arc length equals the radius. A full circle is 2π radians.

Radians provide a natural way of angle measurement for mathematical calculations, particularly those involving calculus and trigonometry. Converting between these units is an essential skill for students, making radian conversion an important concept to grasp.

Radian Conversion

Converting between degrees and radians is crucial in mathematics, especially in trigonometry and calculus. The formula used for conversion is the ratio \left\( \frac{\pi}{180} \right\) radians per degree. To convert an angle from degrees to radians, multiply the number of degrees by this ratio.

Let's illustrate how this works with an example:

• Given angle: \(-405^\circ\)
• Conversion formula: angle in radians = degrees × \left\( \frac{\pi}{180} \right\)
• Substitute and simplify: \(-405 \times \frac{\pi}{180} = -\frac{9\pi}{4} \) radians

This process shows that angle measurements can be seamlessly transitioned between degrees and radians. Understanding this conversion helps you apply trigonometric functions more effectively in varied applications.