Let θ be a positive angle. If the number of degrees in θ is divided by the number of radians in θ, then an irrational number 180 / π results. If the number of degrees in θ is multiplied by the number of radians in θ, then an irrational number 125π / 9 results. The angle θ must be equal to
50°
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 $R$:
$$ D = R \times \frac{180}{\pi} $$Now, substitute this expression for $D$ into equation (2):
$$ \left( R \times \frac{180}{\pi} \right) \times R = \frac{125\pi}{9} $$ $$ \frac{180}{\pi} \times R^2 = \frac{125\pi}{9} $$To find $R^2$, we can multiply both sides by $\frac{\pi}{180}$:
$$ R^2 = \frac{125\pi}{9} \times \frac{\pi}{180} $$ $$ R^2 = \frac{125 \times \pi^2}{9 \times 180} $$ $$ R^2 = \frac{125 \pi^2}{1620} $$We can simplify the fraction $\frac{125}{1620}$ by dividing both numerator and denominator by their greatest common divisor, which is 5:
$$ \frac{125}{5} = 25 $$ $$ \frac{1620}{5} = 324 $$So the equation for $R^2$ becomes:
$$ R^2 = \frac{25 \pi^2}{324} $$Since $\theta$ is a positive angle, $R$ must be positive. Taking the square root of both sides:
$$ R = \sqrt{\frac{25 \pi^2}{324}} $$ $$ R = \frac{\sqrt{25} \times \sqrt{\pi^2}}{\sqrt{324}} $$ $$ R = \frac{5 \times \pi}{18} $$ $$ R = \frac{5\pi}{18} \text{ radians} $$Finding the Angle in Degrees
Now that we have the value of $R$, we can find the value of $D$ using the relationship $D = R \times \frac{180}{\pi}$:
$$ D = \frac{5\pi}{18} \times \frac{180}{\pi} $$ $$ D = \frac{5\pi \times 180}{18 \times \pi} $$Cancel out the common terms:
$$ D = \frac{5 \times 180}{18} $$ $$ D = 5 \times \frac{180}{18} $$ $$ D = 5 \times 10 $$ $$ D = 50 \text{ degrees} $$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
| Concept | Description | Formula |
|---|---|---|
| Angle in Degrees ($D$) | A unit of angle measurement, where a full circle is 360 degrees. | |
| Angle in Radians ($R$) | A unit of angle measurement based on the arc length of a circle, where a full circle is $2\pi$ radians. | |
| Degrees to Radians Conversion | Formula to convert an angle measure from degrees to radians. | $R = D \times \frac{\pi}{180^\circ}$ |
| Radians to Degrees Conversion | Formula to convert an angle measure from radians to degrees. | $D = R \times \frac{180^\circ}{\pi}$ |
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 $\frac{180^\circ}{\pi \text{ radians}}$.
Important Questions from Trigonometry
The given equation can be reduced to
If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?
What is sin 2α equal to?
If \(\sin θ = \frac{8}{{17}}\) , then find the value of tan θ.
If cos(A - B) = \(\frac{\sqrt 3}{2}\) and cot(A + B) = \(\frac{1}{\sqrt 3}\) , Where A - B and A + B are acute angles, then (2A - 3B) is equal to:
