Problem 72 - Precalculus: A Concise Course

Question

Convert the angle measure from degrees to radians. Round to three decimal places. $$ 0.54^{\circ} $$

Step-by-Step Solution

Answer

The angle \(0.54^{\circ}\) is approximately \(0.009\) radians when rounded to three decimal places.

Step 1: Identify the Given Angle in Degrees and the Conversion Formula

The angle given in degrees is \(0.54^{\circ}\). Utilize the formula for converting degrees to radians: \[ Radian = Degree × \frac{π}{180} \].

Step 2: Substituting degree into the Formula

Substitute the given degree (0.54) into the formula: \[ Radian = 0.54 × \frac{π}{180} \].

Step 3: Calculation and Rounding off

Perform the multiplication and round off the result to 3 decimal places.

Key Concepts

Degrees to Radians

Converting degrees to radians is a common task in mathematics, especially in trigonometry, calculus, and physics. To make this conversion, we use the relationship between the two units. A circle has 360 degrees and also equals to \(2\pi\) radians. Therefore,

• 1 degree is equivalent to \(\frac{\pi}{180}\) radians

By applying this conversion factor, we can easily convert any angle given in degrees into radians by multiplying the degree value by \(\frac{\pi}{180}\). For example, to convert \(0.54^{\circ}\) into radians, we multiply:

• \(0.54 \times \frac{\pi}{180} = 0.00942 \text{ radians (rounded to 3 decimal places)}\)

Understanding this formula allows you to move back and forth between degrees and radians effectively, aiding in solving a variety of mathematical problems.

Angle Measurement

Angles can be measured in different units, with degrees and radians being the most common. Every angle consists of two rays that emanate from a single point called the vertex.

• Degrees are one way to measure an angle, often useful in practical everyday settings.

On the other hand, radians are often more practical in higher mathematics, especially calculus, because of their natural relationship with the unit circle.
The choice of unit often depends on the context:

• For navigation, crafts, or architecture, degrees are more intuitive.
• In mathematics or physics, radians provide a more seamless approach.

By understanding the different units of angle measurement, students can be better prepared to tackle problems in both standard and academic contexts.

Radians

Radians provide an elegant way to express angles. They link directly to the geometry of a circle. One of the beauties of radians is the simplicity with which they define angles in relation to a circle's radius.

• A full circle measures \(2\pi\) radians, as this equals the circumference of a unit circle.
• A half circle is \(\pi\) radians, while a quarter circle is \(\frac{\pi}{2}\) radians.

The radian measure comes in handy particularly for calculations involving trigonometric functions, where many formulas are simpler when using radians instead of degrees.
When first learning about radians, students might find it helpful to visualize the conversion process and how it translates angular measures into a more algebraically convenient form. Becoming comfortable with radians opens up many analytical conveniences in mathematics and physics.