Problem 21 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ -1.2 $$

Step-by-Step Solution

Answer

-68.75°

Step 1: Understand the relationship

To convert radians to degrees, we need to use the conversion factor between radians and degrees. Remember that \( 2\pi \) radians is equivalent to \( 360^\circ \). So, \( 1\) radian is equivalent to \( \frac{360^\circ}{2\pi} \).

Step 2: Set up the conversion equation

To convert \(-1.2 \) radians to degrees, we use the formula: \( \text{degrees} = \text{radians} \times \frac{180^\circ}{\pi} \). So, substitute \(-1.2\) into the formula: \( \text{degrees} = -1.2 \times \frac{180^\circ}{\pi} \).

Step 3: Calculate the value

Calculate \(-1.2 \times \frac{180}{\pi}\) by substituting \( \pi \approx 3.14159 \). Thus, \( \text{degrees} \approx -1.2 \times 57.2958 \).

Step 4: Simplify the expression

Perform the multiplication: \(-1.2 \times 57.2958 \approx -68.75496 \).

Step 5: State the final answer

Round \(-68.75496\) to two decimal places to obtain \(-68.75^\circ\).

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry and trigonometry. Understanding how to measure angles helps us describe the rotation from one ray to another around a common vertex. In simple terms, angle measurement is about quantifying how much one line or plane is tilted compared to another.

There are two primary units used for measuring angles: degrees and radians. Each system has its own way of expressing the size of an angle, and it is crucial to know how to convert between them to solve problems that require different units.

• Degrees: A degree is \(\frac{1}{360}\) of a full circle. This means a full circle is 360 degrees, making it easy to understand and commonly used in everyday contexts like geometry and navigation.
• Radians: A radian is another way to measure angles, derived from the radius of a circle. One radian is the angle created when the radius is wrapped along the circumference of the circle. It's often used in calculus and advanced mathematics.

Understanding both systems allows for flexibility in solving mathematical and real-world problems. It provides a deeper comprehension of how angles can be represented in different contexts.

Degrees

Degrees are a familiar measure for most of us, commonly used to describe angles in geometry, navigation, and various applications. One complete revolution around a circle is divided into 360 equal parts, known as degrees.

This system is convenient because of the divisibility of the number 360, which has many divisors, making it easier to work with common geometric figures:

• A right angle is 90 degrees.
• A straight line forms a 180-degree angle.
• A full circle encompasses 360 degrees.

Degrees are often represented with the degree symbol (\( ^\circ \)), making it easy to recognize. When converting angles from radians to degrees or vice versa, using the conversion factor \( \frac{180^\circ}{\pi} \) is essential. Mastering these conversions enables smoother mathematical transition between different types of problems.

Radians

Radians provide a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the circle's radius and arc length. There are \(2\pi\) radians in a full circle, which simplifies many mathematical equations and is particularly useful in calculus.

To understand radians, imagine a circle with a radius and see how the radius stretches along the circumference. The length of the arc is equal to the radius when the angle carries a measure of one radian. Hence, radians naturally integrate with the geometry of a circle, making calculations involving angular motion straightforward.

• Full Circle: \(2\pi\) radians
• Straight Line: \(\pi\) radians
• Right Angle: \(\frac{\pi}{2}\) radians

Conversions between radians and degrees are necessary because different problems require different units. For instance, in the given exercise, converting \-1.2\ radians to degrees involves multiplying by \(\frac{180^\circ}{\pi}\), which relies on the fundamental relationship between these two measurements.