Problem 19 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ -1.2 $$

Step-by-Step Solution

Answer

-68.75 degrees.

Step 1: Understand the conversion formula

To convert radians into degrees, use the formula: Degrees = Radians × (180/π). This is because π radians is equal to 180 degrees.

Step 2: Substitute the radian value into the formula

Substitute -1.2 radians into the conversion formula. So, we have Degrees = -1.2 × (180/π).

Step 3: Calculate the conversion

First, calculate 180/π, which is approximately 57.2958. Then multiply this result by -1.2: Degrees = -1.2 × 57.2958 ≈ -68.75496.

Step 4: Round the result

Since degree measures are usually rounded to one or two decimal places, round -68.75496 to either -68.75 or -68.76, depending on the needed precision. Here, round to two decimal places: -68.75.

Key Concepts

Radian to Degree Conversion

Radian to degree conversion is an essential skill in trigonometry and angle measurement.When we measure angles, we often use different units; radians and degrees are the most common. To convert between these units, a simple formula is used.A complete circle is divided into 360 degrees or, equivalently, \(2\pi\) radians. This leads us to a helpful conversion factor: \(\pi = 180^{\circ}\).Given an angle in radians, say \(-1.2\), the conversion process involves multiplying by \(\frac{180}{\pi}\). Using this, the formula for converting radians to degrees can be written as:

• Degrees = Radians × (180/π).

This formula allows us to switch smoothly between these units, opening up various possibilities in solving trigonometric problems.This basic understanding of radian to degree conversion is crucial, especially when dealing with complex trigonometric calculations in real-world and theoretical scenarios.

Angle Measurement

Angle measurement is a fundamental aspect in geometry, physics, and trigonometry. It provides a way to quantify the size of an angle, essentially helping us understand the shape and direction of lines and planes.Angles can be measured in different units, such as degrees or radians:

• Degrees are more intuitive for everyday use. A full circle is 360 degrees, making it easy to divide into standard, commonly-used angles like 90° or 45°.
• Radians offer a more mathematical perspective, especially useful in trigonometry. One full rotation (360°) is equal to \(2\pi\) radians. This is incredibly useful as it relates directly to the properties of circles.

Using the appropriate unit for measuring angles depends on the context.Some situations rely heavily on degrees for mapping and navigation, whereas radians are preferred in mathematical functions and trigonometric calculations.Understanding different angle measurements helps create a flexible approach to problem-solving.It ensures that we can transition seamlessly between geometrical intuition and mathematical precision.

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. This study covers more than just triangular shapes; it delves into the wave-like properties of angles, becoming essential in fields like physics, engineering, and beyond. Key trigonometric principles include:

• Trigonometric Ratios: These are the sine, cosine, and tangent functions, which relate the angles of a triangle to the lengths of its sides.
• Unit Circle: A concept in trigonometry where angles are measured, and trigonometric values are derived from a circle with a radius of one.
• Angle Conversion: Often in trigonometric problems, angles need converting between degrees and radians to utilize formulas and solve equations.

Trigonometry is also renowned for its identities and theorems, like the Pythagorean identity and laws of sines and cosines. These principles underpin a wide range of applications, from calculating heights and distances to understanding sound and light waves. Understanding the basics of trigonometry, including angle conversions, offers a comprehensive view of how angles interact in both theoretical and practical settings.