Question

Convert the following angles in degrees to radians: (a) \(12^{\circ}\) (b) \(65^{\circ}\) (c) \(200^{\circ}\) (d) \(340^{\circ}\) (e) \(1000^{\circ}\)

Short answer

Question: Convert the following angles from degrees to radians: (a) \(12^{\circ}\) (b) \(65^{\circ}\) (c) \(200^{\circ}\) (d) \(340^{\circ}\) (e) \(1000^{\circ}\) Answer: (a) \(\frac{\pi}{15}\) radians (b) \(\frac{13\pi}{36}\) radians (c) \(\frac{10\pi}{9}\) radians (d) \(\frac{17\pi}{9}\) radians (e) \(\frac{50\pi}{9}\) radians

Step-by-step solution

(a) Convert \(12^{\circ}\) to radians

Firstly, let's use the formula to convert \(12^{\circ}\) to radians: Radians = (\(12 * \pi\)) / 180 Radians = \(\frac{12\pi}{180} = \frac{\pi}{15}\) So, \(12^{\circ}\) is equal to \(\frac{\pi}{15}\) radians.

(b) Convert \(65^{\circ}\) to radians

Now, we'll convert \(65^{\circ}\) to radians using the same formula: Radians = (\(65 * \pi\)) / 180 Radians = \(\frac{65\pi}{180} = \frac{13\pi}{36}\) So, \(65^{\circ}\) is equal to \(\frac{13\pi}{36}\) radians.

(c) Convert \(200^{\circ}\) to radians

Next, let's convert \(200^{\circ}\) to radians: Radians = (\(200 * \pi\)) / 180 Radians = \(\frac{200\pi}{180} = \frac{10\pi}{9}\) So, \(200^{\circ}\) is equal to \(\frac{10\pi}{9}\) radians.

(d) Convert \(340^{\circ}\) to radians

Now, let's convert \(340^{\circ}\) to radians: Radians = (\(340 * \pi\)) / 180 Radians = \(\frac{340\pi}{180} = \frac{17\pi}{9}\) So, \(340^{\circ}\) is equal to \(\frac{17\pi}{9}\) radians.

(e) Convert \(1000^{\circ}\) to radians

Finally, let's convert \(1000^{\circ}\) to radians: Radians = (\(1000 * \pi\)) / 180 Radians = \(\frac{1000\pi}{180} = \frac{50\pi}{9}\) So, \(1000^{\circ}\) is equal to \(\frac{50\pi}{9}\) radians.

Key concepts

Radian Measure

When studying angles, one of the fundamental concepts you'll encounter is the radian measure. It's often considered the standard unit of angular measure used in many areas of mathematics, particularly trigonometry. A radian measures an angle at the center of a circle that is subtended by an arc whose length is equal to the radius of the circle.

In simpler terms, imagine you're unwrapping your circle from the circumference and laying it straight. The length of a section of the circumference that is the same length as the radius of that circle would be an angle of one radian. For a complete turn around a circle, which is 360 degrees, there are roughly 6.283 radians, equivalent to \(2\text{π}\text{ radians}\).

This leads to a powerful understanding when working with circles and angles: the circumference of a circle can also be described as \(2\text{π}r\), where \(r\) is the radius of the circle. If you were to compare degrees and radians, you'd find that 180 degrees equal to \(\text{π}\text{ radians}\), thus creating a simple proportional relationship between degrees and radians which can be used to convert from one unit to another. As you delve deeper into trigonometry and calculus, you'll encounter radians more frequently because they provide a natural way for measuring angles in terms of the radii of circles.

Degree to Radian Formula

Converting degrees to radians is a common necessity in trigonometry, and luckily, there's a straightforward formula to make this switch. The degree to radian conversion formula arises from the proportion \(180^\circ : \text{π} \text{ radians}\).

To convert an angle from degrees to radians, you use the formula:

\[ \text{Radians} = \frac{\text{Degree}}{180} \times \text{π} \]
Using this relationship allows you to quickly and easily translate between the two systems of measuring angles. For example, to convert 45 degrees into radians:

\[ \text{Radians} = \frac{45}{180} \times \text{π} = \frac{\text{π}}{4} \]
It's a key formula for students to memorize or understand, not just for homework but also when applying trigonometry concepts to real-world problems, such as in physics and engineering. Moreover, when angles are represented in radians, it simplifies the understanding and calculation of other trigonometric functions, aiding in the students' overall comprehension of the subject.

Trigonometry

Trigonometry is the branch of mathematics that deals primarily with triangles, circles, waves, and oscillations. It's rooted deeply in the study of geometric figures and seeks to understand the relationships between side lengths and angles of triangles. More broadly, it encompasses the study of the properties of any angle, be they acute or obtuse, and the functions that describe those relationships, like sine, cosine, and tangent.

Key to trigonometry are the trigonometric functions, which are defined for angles, which could either be measured in degrees or radians. These functions are essential tools for solving problems in physics, engineering, and computer science, particularly when describing periodic phenomena.

For instance, when you consider the motion of a pendulum or the sound waves produced by a musical instrument, trigonometric functions can model these periodic behaviors precisely. Understanding how to convert degrees to radians, therefore, is not just a mathematical exercise—it's a foundational skill that's critical for application in a wide variety of scientific and technical fields, ultimately facilitating the deep exploration of the world around us.