Problem 11 - Algebra and Trigonometry

Question

\(1-12\) . Find the radian measure of the angle with the given degree measure. $$ 7.5^{\circ} $$

Step-by-Step Solution

Answer

The radian measure of the angle is \(\frac{\pi}{24}\).

Step 1: Understanding Degree to Radian Conversion

To convert an angle from degrees to radians, we use the conversion formula:\[\text{{Radians}} = \text{{Degrees}} \times \frac{\pi}{180}\]

Step 2: Plug the Degree Measure into the Formula

Given the angle measure is \(7.5^{\circ}\), plug this value into the conversion formula:\[\text{{Radians}} = 7.5 \times \frac{\pi}{180}\]

Step 3: Simplify the Expression

Simplify the expression by multiplying \(7.5\) by \(\frac{\pi}{180}\):\[\text{{Radians}} = \frac{7.5\pi}{180} = \frac{\pi}{24}\]

Step 4: Write the Final Radian Measure

Thus, the radian measure of the angle when the degree measure is \(7.5^{\circ}\) is \(\frac{\pi}{24}\).

Key Concepts

Degree to Radian Conversion

Converting degrees to radians is essential in the world of mathematics, especially in trigonometry. The main idea behind this conversion is to change one way of measuring angles to another, more standardized form used in calculus and advanced mathematics.
The formula is quite simple: multiply the degree measure by \( \frac{\pi}{180} \).

• This conversion factor \(\frac{\pi}{180}\) comes from the fact that a full circle in degrees is \(360^{\circ}\) and a full circle in radians is \(2 \pi\).
• Thus, \(360^{\circ}\) is equivalent to \(2\pi\) radians, which simplifies to \(\frac{\pi}{180}\) per degree.

For example, converting \(7.5^{\circ}\) to radians involves multiplying by this factor: \(7.5 \times \frac{\pi}{180}\). The result is \(\frac{\pi}{24}\), a more precise measure in terms of \(\pi\).

Angle Measurement

Understanding how angles are measured is pivotal in math and science. Angles can be measured in different units, the most common being degrees and radians.

• Degrees are a more intuitive measure for simple geometrical problems and are widely taught in early education.
• Radians, however, provide a more coherent system for advanced applications, like calculus, because they relate directly to the circumference of a circle.

An angle that covers a full circle measures \(360^{\circ}\) in degrees or \(2\pi\) radians. Half a circle, or a straight angle, is \(180^{\circ}\) or \(\pi\) radians.
Understanding how these units interrelate helps when transitioning from geometric concepts to algebraic and calculus applications.

Trigonometry Basics

Trigonometry is fundamental for understanding angles and their relationships in various geometric figures. Here are some core principles:

• Trigonometry focuses on the properties of triangles, especially right triangles.
• The primary trigonometric functions are sine, cosine, and tangent, which relate to the angles and sides of a triangle.

These functions are often represented in radians because it simplifies many formulae and calculations. For instance, the sine function is periodic with a period of \(2\pi\), meaning that its values repeat every \(2\pi\) radians. This understanding becomes crucial in advanced math topics, such as wave functions and oscillations.
By mastering the basics of radians and trigonometry, many other mathematical concepts become easier to grasp.