Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-46.52^{\circ}$$

Step-by-Step Solution

Answer

The angle in radians, rounded to three decimal places, is \(-0.812\) radians.

Step 1: Identify the degree measure

The given angle is \(-46.52^{\circ}\).

Step 2: Convert degrees to radians

To convert the angle from degrees to radians, use the relation that \(\pi\) radians equals \(180^{\circ}\). Therefore, multiply the angle by \(\frac{\pi}{180}\). This gives: \(-46.52 \times \frac{\pi}{180}\).

Step 3: Compute the result

After doing the multiplication, round off the result to three decimal places.

Key Concepts

Angle Measurement Conversion

In mathematics, converting the measurement of an angle from one unit to another is a fundamental skill, especially for students studying geometry and trigonometry. Most commonly, angles are measured in degrees or radians. To perform a conversion, we apply a specific mathematical relationship between these two units.

The key to understanding this process lies in knowing that the circle is divided into 360 degrees, which is equivalent to \(2\pi\) radians. From this relationship, we can deduce that \(180^\circ\) is equal to \(\pi\) radians. Therefore, to convert degrees to radians, we multiply the degree measurement by the fraction \(\frac{\pi}{180}\). Conversely, to convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

For example, let's convert \-46.52^\circ\ to radians. We follow these steps:

• Multiply \-46.52\ by \(\frac{\pi}{180}\).
• The result from the multiplication is the angle in radians.

Radian

A radian is a unit of angular measure used in many areas of mathematics, especially in trigonometry. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. This is an extremely important concept because it provides a direct relationship between the linear and angular measurements of a circle.

The circumference of a circle is \(2\pi r\), where \(r\) is the radius. Since the length of the circumference is one full rotation around the circle, and this is \(2\pi\) times the radius, it follows that a full circle is \(2\pi\) radians. This realization allows for trigonometric functions to be defined in terms of radians, making calculations involving arcs and angles much more straightforward in many cases.

Degree

The degree is a measure of angle commonly used in several different contexts, such as in geography for measuring latitude and longitude, and in various branches of science. One degree is defined as \(\frac{1}{360}\) of a full rotation. The concept of measuring angles in degrees originated from the ancient Babylonians, who devised a base-60 number system. A full circle was divided into 360 degrees because the number 360 has many divisors, making it easier to work with fractions.

When working in degrees, it's important to be comfortable with fractions and have an understanding of how to work with degree-minute-second notation, as angles are often presented in this way, especially in navigation and surveying.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's broadly used in various scientific and engineering fields, such as physics, architecture, and even computer graphics. Key to trigonometry are the sine, cosine, and tangent functions, which relate specific angles in a right triangle to ratios of the sides of the triangle.

The use of radians in trigonometry is especially valuable as it simplifies the relationship between angular motion and linear distance, such as in the calculations of arc lengths or the areas of sectors in circles. In advanced mathematics, radians are often the preferred unit because they allow for the easy integration and differentiation of trigonometric functions, essential in calculus.