Solving the Angle Measurement Problem

The question provides us with information about a positive angle, let's call it \(\theta\). We are given two relationships involving the measure of this angle in degrees and radians. Let \(D\) represent the measure of angle \(\theta\) in degrees, and \(R\) represent the measure of angle \(\theta\) in radians.

Setting Up the Equations

According to the problem statement:

• The number of degrees in \(\theta\) divided by the number of radians in \(\theta\) is equal to \(\frac{180}{\pi}\). This can be written as: \( \frac{D}{R} = \frac{180}{\pi} \quad (1) \)
• The number of degrees in \(\theta\) multiplied by the number of radians in \(\theta\) is equal to \(\frac{125\pi}{9}\). This can be written as: \( D \times R = \frac{125\pi}{9} \quad (2) \)

We now have a system of two equations with two variables, \(D\) and \(R\). We need to find the value of \(D\), which is the angle \(\theta\) in degrees.

Solving the System of Equations

Let's solve this system. From equation (1), we can express \(D\) in terms of \(R\):

Now, substitute this expression for \(D\) into equation (2):

To find \(R^2\), we can multiply both sides by \(\frac{\pi}{180}\):

We can simplify the fraction \(\frac{125}{1620}\) by dividing both numerator and denominator by their greatest common divisor, which is 5:

So the equation for \(R^2\) becomes:

Since \(\theta\) is a positive angle, \(R\) must be positive. Taking the square root of both sides:

Finding the Angle in Degrees

Now that we have the value of \(R\), we can find the value of \(D\) using the relationship \(D = R \times \frac{180}{\pi}\):

Cancel out the common terms:

Thus, the angle \(\theta\) is equal to \(50^{\circ}\).

Verification of the Solution

Let's check if \(D=50\) and \(R = \frac{5\pi}{18}\) satisfy the original equations:

• Equation (1): \(\frac{D}{R} = \frac{50}{\frac{5\pi}{18}} = 50 \times \frac{18}{5\pi} = \frac{50}{5} \times \frac{18}{\pi} = 10 \times \frac{18}{\pi} = \frac{180}{\pi}\). This is correct.
• Equation (2): \(D \times R = 50 \times \frac{5\pi}{18} = \frac{50 \times 5\pi}{18} = \frac{250\pi}{18}\). Simplifying the fraction \(\frac{250}{18}\) by dividing both by 2 gives \(\frac{125}{9}\). So, \(D \times R = \frac{125\pi}{9}\). This is also correct.

Both conditions are satisfied, confirming that the angle is \(50^{\circ}\).

Revision Table: Angle Degrees and Radians

Additional Information: Angle Measurement Units

Angles can be measured using different units, the most common being degrees and radians. Understanding the relationship between these units is fundamental in trigonometry and many areas of physics and engineering.

• Degrees: This unit is widely used in geometry and navigation. A full circle is divided into 360 equal parts, with each part being one degree (\(1^\circ\)). This system dates back to ancient times, possibly related to the number of days in a year.
• Radians: This unit is based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. This unit is often preferred in calculus and theoretical mathematics because it leads to simpler formulas. A full circle is \(2\pi\) radians.

The conversion factor between degrees and radians comes from the fact that \(180^\circ\) is equal to \(\pi\) radians. This gives us the conversion ratios \(\frac{\pi \text{ radians}}{180^\circ}\) and \(\frac{180^\circ}{\pi \text{ radians}}\).