In Exercises 13 to 18, classify each angle by quadrant, and...

In Exercises 13 to $18,$ classify each angle by quadrant, and state the measure of the positive angle with measure less than $360^{\circ}$ that is coterminal with the given angle.
$$\alpha=2456^{\circ}$$

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. They can be found by adding or subtracting full rotations, typically 360° in degree measure, to a given angle. This concept is fundamental in trigonometry because many trigonometric functions are periodic, meaning that angles differing by a full rotation have the same sine, cosine, and tangent values.

Angle Reduction Using Modular Arithmetic

Angle reduction involves finding an equivalent angle within a standard interval (usually between 0° and 360°) that has the same terminal side as the original angle. This is typically achieved using modular arithmetic, by reducing the given angle modulo 360°. The process simplifies calculations and helps in analyzing the positions and properties of the angle on the unit circle.

Quadrant Classification

Quadrant classification is the process of determining in which of the four quadrants an angle’s terminal side falls when drawn in standard position. The four quadrants are defined by the positive and negative directions of the x- and y-axes, and each quadrant corresponds to a specific range of angle measures. This classification is essential for understanding the signs of trigonometric functions and the geometric relationships of angles on the coordinate plane.

Transcript

00:03 In problem number 17, we are told that alpha is the angle 2 ,456 degrees.

00:11 That's abnormally large, and in my opinion, it's a little overkill for what they're trying to accomplish here, but that's okay.

00:18 We can get through it, no problem.

00:21 In fact, i like to take a little bit extra time and do it visually because i think that benefits a lot of learners.

00:28 The fact of the matter is the problems are just basic arithmetic.

00:32 You're just adding or subtracting multiples of 360 depending on what your angle is.

00:37 So let's try to do it visually.

00:40 Maybe that'll improve understanding a little bit.

00:43 So anyway, we want to figure out in what quadrant does the terminal side of that angle end up? and then what is the equivalent co -terminal angle within the first revolution of the circle? so from zero to 360 degrees.

01:00 So literally, if you wanted to do it arithmetic -wise, you would take 2456 and then you would subtract 360 and then subtract 360 again and subtract 360 again until you get into that range of zero to 360.

01:17 And then that's going to be your answer.

01:19 If you want to do it that way, i like to do it visually.

01:24 That's just me.

01:26 All right.

01:26 So let's figure out where this angle of 2 ,000.

01:31 456 degrees is first.

01:34 So we're going to use lots and multiples of 360, right? hopefully i can do the arithmetic without too much difficult.

01:43 So one time around the circle, that's 360.

01:48 Two times around the circle, that's going to be 720...