Problem 19 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ 412^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle between 0 and 360 degrees for 412 degrees is 52 degrees.

Step 1: Identify the Given Angle

The given angle is 412 degrees. We need to find a coterminal angle that falls within the range of 0 to 360 degrees.

Step 2: Understand Coterminal Angles

Coterminal angles are angles that have the same initial and terminal sides. To find a coterminal angle, you can add or subtract full rotations (multiples of 360 degrees) from the given angle.

Step 3: Subtract Full Rotations

Since 412 is larger than 360, we need to subtract 360 to bring it within the desired range. Calculate: \[ 412 - 360 = 52 \]

Step 4: Check the Result

The result of 52 degrees is a coterminal angle with the original 412 degrees. Verify that it lies within the required range, 0 to 360 degrees. Since 52 falls within this range, it is the solution.

Key Concepts

Angle Measurement

When discussing angles, we often refer to their measurement, which indicates the amount of rotation from the initial side to the terminal side. Angles are commonly measured in degrees. One full circle equates to 360 degrees. This means if you start at 0 degrees and make a complete loop back to your starting point, you've covered 360 degrees. Angles can be greater than 360 degrees or even negative, depending on the direction and number of rotations.

For example, an angle measuring 412 degrees means the terminal side has rotated once fully (360 degrees) and continued an additional 52 degrees past the point of a complete circle. Understanding this concept is essential for identifying coterminal angles, which can be larger or smaller than the 'standard' 0 to 360-degree range.

Subtracting Full Rotations

To find a coterminal angle within a specific range, such as 0 to 360 degrees, it helps to use the idea of full rotations. Subtracting full rotations involves removing multiples of 360 degrees from an angle that exceeds 360 degrees. This process brings the angle down to its simplest form, as defined within one full circle or rotation.

For example, when dealing with an angle of 412 degrees, you subtract 360 degrees, a full rotation, to simplify the angle to 52 degrees. In mathematical terms: \[ 412 - 360 = 52 \]This simplified angle of 52 degrees shares the same direction and position as the original angle, thereby making them coterminal. Through this method, any angle can be reduced to lie within the more manageable range of 0 to 360 degrees if necessary.

Range of Angles

The range of angles is a critical concept in trigonometry and geometry. When it comes to coterminal angles, the most typical range considered is from 0 to 360 degrees. This is because angles repeat after a full rotation. When considering calculations and problem-solving, getting angles within this range simplifies comparison and understanding.

Any angle that falls outside of the 0 to 360-degree range can be modified by either adding or subtracting 360 degrees incrementally until it fits into this range. For example, 412 degrees was adjusted by subtracting a full rotation of 360 degrees, bringing it into the desired range with the final result of 52 degrees. This practice helps in identifying angles with the same terminal side, making them easier to work with and understand in practical scenarios like navigation, engineering, or physics.