Problem 47 - Algebra and Trigonometry

Question

\(43-48=\) Find an angle between 0 and 2\(\pi\) that is coterminal with the given angle. $$ \frac{17 \pi}{4} $$

Step-by-Step Solution

Answer

The angle \(\frac{\pi}{4}\) is coterminal with \(\frac{17\pi}{4}\) and lies between 0 and 2\(\pi\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find a coterminal angle within a specific range, you adjust the angle by adding or subtracting full rotations (in radians, this is \(2\pi\)).

Step 2: Convert the Angle to a Usable Form

The given angle is \(\frac{17\pi}{4}\). We'll focus on expressing this angle in terms of complete revolutions (multiples of \(2\pi\)).Calculate \(\frac{17\pi}{4} \, \text{mod} \, 2\pi\):First, convert \(2\pi\) to a fraction with the same denominator: \(2\pi = \frac{8\pi}{4}\).Now, calculate \(\frac{17\pi}{4} - n\cdot\frac{8\pi}{4}\) to bring it into the desired range.

Step 3: Simplify to Find the Coterminal Angle

Subtract the full rotations (multiples of \(8\pi/4\)):\[\frac{17\pi}{4} - \frac{16\pi}{4} = \frac{\pi}{4}\]The angle \(\frac{\pi}{4}\) is between \(0\) and \(2\pi\).

Step 4: Verify the Solution

Verify that the resulting angle, \(\frac{\pi}{4}\), is indeed coterminal with the given angle within the specified range. This angle meets the requirement of being between \(0\) and \(2\pi\).

Key Concepts

Radians

When measuring angles in terms of curvature instead of degrees, we use the unit called radians. One radian is defined as the angle created when the radius of a circle is wrapped along the circle's edge. In simpler terms, if you take the radius and lay it around the circle's circumference, the angle at the center is 1 radian. The entire circumference of a circle is described by the angle of \[2\pi ?\approx? 6.28\]radians, because the circumference is approximately 6.28 times the length of the radius.

• Radians provide a natural way to describe angles for circular and rotational movements.
• This unit is incredibly important in calculus and physics, where rotational properties are analyzed.

Understanding radians will help in easily converting between angles and understanding real-world periodic scenarios.

Angle Conversion

Converting angles is essential when working between different units or simplifying angles for calculation. Angles can be expressed in degrees, which are more commonly used in everyday settings, or radians, which are crucial in advanced mathematics and physics. For conversion, keep these important relationships in mind:

• 1 full circle = 360 degrees = \2\pi\ radians
• 1 radian = \(\frac{180}{\pi}\) degrees
• 1 degree = \(\frac{\pi}{180}\) radians

When converting from radians to degrees or vice versa, we adapt these relationships accordingly. For instance, to convert a given angle in radians to degrees, multiply by \[\frac{180}{\pi}\]and to go from degrees to radians, multiply by \[\frac{\pi}{180}\].By understanding angle conversion, you can handle different problems requiring changes in units confidently.

Complete Revolutions

When finding coterminal angles, understanding complete revolutions is a vital concept. A complete revolution around a circle returns you to your starting point, which translates to \[2\pi\]radians or 360 degrees. Each complete revolution involves moving through an entire circle once.To find coterminal angles:

• Add or subtract multiples of \2\pi\ radians to or from your initial angle.
• This adjusts the angle's measure by entire circle rotations, affecting the angle's position but not its terminal side.

For example, in the step-by-step solution above, we computed\(\frac{17\pi}{4} - \frac{8\pi}{4} - \frac{8\pi}{4}\)to identify a coterminal angle.Finding coterminal angles by considering complete revolutions is fundamental for understanding periodic behaviors in trigonometry and their applications in real-world phenomenons like waves and oscillations.