Problem 77 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$-\frac{313 \pi}{9}$$

Step-by-Step Solution

Answer

\(\frac{11\pi}{9}\)

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial side and terminal side but may have different measures. To find a coterminal angle, you can add or subtract full circles (in radians, this is equivalent to \(2\pi\)) from the given angle until you find the smallest positive angle.

Step 2: Add or Subtract Full Radians

The given angle is \(-\frac{313\pi}{9}\). To find a coterminal angle, we can keep adding \(2\pi\) (or \(\frac{18\pi}{9}\) to match denominators) until the angle is positive. Let's start with: if \(-\frac{313\pi}{9}\) + \(\frac{324\pi}{9}\) = \(\frac{11\pi}{9}\).

Step 3: Verify the Calculated Angle

Now that we have \(\frac{11\pi}{9}\), we need to ensure it's between 0 and \(2\pi\). Since \(0 < \frac{11\pi}{9} < 2\pi\), \(\frac{11\pi}{9}\) is the smallest positive angle coterminal with the original angle.

Key Concepts

Radian Measure

Radian measure is a way of expressing the size of an angle. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's radius. One radian is the angle created when the arc length is equal to the radius of the circle.
This is pivotal because it interrelates angle measurements directly with the dimensions of a circle.
In practical terms:

• A full circle in radians is equal to \(2\pi\).
• A half circle or a straight angle is \(\pi\) radians.
• For computations, \(180^{\circ}\) is equivalent to \(\pi\) radians.

Understanding radians particularly helps in various fields like calculus and physics, where the natural properties of the circle come into play frequently.
It often provides easier mathematical formulas and calculations compared to degrees.

Positive Angle

When we talk about angles, sometimes they come with a negative sign. This simply means they're measured in the opposite direction, usually clockwise.
However, to find a coterminal positive angle, we need to make this measurement positive and less than a complete circle, in this context, less than \(2\pi\) radians.
For example, if an angle is given in a negative measure such as \(-\frac{313\pi}{9}\), the challenge is to translate this to a positive angle while maintaining it as a coterminal angle.

• Start by considering how many full circles \(2\pi\) fit into the negative angle.
• By adding the appropriate number of \(2\pi\) rotations, ensure that the resulting angle is within the first full rotation of the unit circle (i.e., between \(0\) and \(2\pi\)).

Ultimately, this ensures that the measure of the angle is positive, which simplifies interpretation and further calculations.

Angle Conversion

Converting between degrees and radians, or between different expressions of radians, is a crucial skill in many areas of mathematics. In our problem, converting what might initially seem like a daunting expression, such as \(-\frac{313\pi}{9}\), into a useful angle involves understanding conversion.To convert an angle from degrees to radians:

• Multiply the degree measure by \(\frac{\pi}{180}\).

Conversely, from radians to degrees:

• Multiply the radian measure by \(\frac{180}{\pi}\).

In the case of complex fractions or large numbers, try simplifying the expression or equating denominators where necessary.
For instance, adding multiples of \(2\pi\) (\(\frac{18\pi}{9}\) in our example) helps simplify the problem. This keeps the work and results within a manageable range, making calculations and visualizations more intuitive.