Problem 76 - 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. $$\frac{47 \pi}{7}$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle is \(\frac{5\pi}{7}\).

Step 1: Understanding Coterminal Angles in Radians

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, we can add or subtract multiples of the full circle, which is \(2\pi\) radians.

Step 2: Express the Angle in Terms of \(2\pi\)

The given angle is \(\frac{47\pi}{7}\). To determine how many full circles (\(2\pi\)) we have in this angle, we need to divide the given angle by \(2\pi\). This is equivalent to \(\frac{47\pi}{7}\div 2\pi = \frac{47}{14}\).

Step 3: Determine the Remainder after Complete Circles

Next, consider the quotient \(\frac{47}{14}\). This division gives approximately \(3.357\), meaning the given angle covers slightly more than 3 full circles (or revolutions). The whole number (3) is the number of complete circles, and we are interested in the remaining fraction to find the coterminal angle within the first circle.

Step 4: Calculate the Remainder Angle

Multiply the decimal part (\(0.357\)) of the division by \(2\pi\) to find the remainder: \(0.357 \times 2\pi \approx 0.714\cdot 7 = \frac{5\pi}{7}\).

Step 5: Check the Remainder Angle

Convert back to verify: the calculation implies the coterminal angle smaller than \(2\pi\) is \(\frac{5\pi}{7}\). This is the smallest positive measure coterminal with the original angle.

Key Concepts

Angles in Radians

Angles are often measured in degrees, but in many mathematical contexts, especially in trigonometry and calculus, radians are the preferred unit. A radian measures the angle created by taking the radius of a circle and wrapping it along the circle's edge. This simplicity is why radians are often used in formulas dealing with circular motion and oscillation.

One full rotation around a circle, which is 360 degrees, is equivalent to \(2\pi\) radians. Thus, to convert an angle from degrees to radians, we use the conversion factor \(\pi/180\), meaning that to convert back from radians to degrees, you would multiply by \(180/\pi\).

• Radians simplify the formulas like arc length (\(s = r \cdot \theta\)) and sector area (\(A = \frac{1}{2} r^2 \cdot \theta\)).
• You can easily identify the number of radians in a half-circle (\(\pi\)), quarter-circle (\(\pi/2\)), and other subdivisions.

Knowing that coterminal angles can be found by adding or subtracting \(2\pi\) times integer values helps when trying to understand the impact of these angles in periodic functions and circles.

Full Circle

The concept of a full circle is crucial when dealing with coterminal angles. A full circle allows us to understand one complete rotation bringing us back to the same starting point. In radians, a full circle is measured as \(2\pi\).

When calculating coterminal angles, we consider how many full circles can fit into the given angle to simplify it to a form comparable to other angles between 0 and \(2\pi\). This process helps in determining the smallest positive coterminal angle:

• To determine how many full circles an angle contains, divide the angle by \(2\pi\).
• Extract the integer part from this division to find out the number of full circles.

For example, with an angle of \(\frac{47\pi}{7}\), dividing by \(2\pi\) results in a quotient that tells us our angle wraps around the circle more than three times. Understanding and using full circles is key to simplifying expressions and understanding their periodic nature.

Remainder Angle

Once you've identified how many full circles are in your angle, the next step is to compute the remainder angle. This is the 'leftover' portion when the full circles are subtracted from the original angle. Finding this helps determine the coterminal angle that falls within the first circle, specifically between 0 and \(2\pi\).

To calculate the remainder angle:

• Subtract the number of full circles, calculated in terms of \(2\pi\), from your angle. This means multiplying the fractional part of your division (e.g., \(0.357\) in our example) by \(2\pi\) again.
• Verify the computation by converting it back if necessary to ensure it stays within the intended range.

For instance, in the example, the leftovers after 3 full circles was approximately \(0.357\) of a full circle. Multiplying this by \(2\pi\) gives the smallest positive coterminal angle of \(\frac{5\pi}{7}\). Thus, understanding remainders helps in simplification and in confirming that angles lie appropriately within a defined boundary.