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Chapter 4: Problem 3

Two angles that have the same initial and terminal sides are ______.

Short Answer

Expert verified
Coterminal.

Step by step solution

01

Understand the Problem

In this problem, we're dealing with two angles that share the same initial and terminal sides. This is a key property of a particular type of angles in geometry.
02

Recall Angular Types

There are many types of angles in geometry, including acute, obtuse, right, straight, and full angles among others. But the answer to this question isn't any of these types. What we're interested in are angles that share the same initial and terminal sides, which is characteristic of coterminal angles.
03

Formulate the Answer

Taking into consideration the behavior of these angles, we conclude the answer to the question. Two angles that have the same initial and terminal sides are known as coterminal angles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Coterminal Angles in Geometry
Geometry is a branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, and shapes. In the context of angles, understanding coterminal angles is essential. Coterminal angles are angles that share the same initial and terminal sides, but they may have different magnitudes.

For instance, an angle of 30 degrees and another angle of 390 degrees are coterminal because the terminal side is the same for both when drawn in standard position (starting from the positive x-axis). This happens because the second angle represents a full 360-degree turn plus an additional 30 degrees. Recognizing coterminal angles is particularly useful in trigonometry where multiple rotations around a circle are considered.
Types of Angles and Their Properties
Geometry classifies angles based on their measure. Here are a few basic types:
  • Acute Angle: Less than 90 degrees.
  • Right Angle: Exactly 90 degrees.
  • Obtuse Angle: Greater than 90 degrees but less than 180 degrees.
  • Straight Angle: Exactly 180 degrees.
  • Reflex Angle: Greater than 180 degrees but less than 360 degrees.
  • Full Rotation: A 360-degree angle.
Coterminal angles could be of any type depending on their measure, but what sets them apart is that their starting point (initial side) and their endpoint (terminal side) are identical in the plane.
Initial and Terminal Sides of Angles
In geometry, every angle has two important components: the initial side and the terminal side. The initial side is where the angle starts, often coinciding with the positive x-axis in a coordinate plane scenario. The terminal side is where the angle ends after rotation. By definition, coterminal angles always have an identical starting and ending position despite the amount of rotation.

To visualize this, imagine a clock with its hands starting from 12 o'clock (the initial side). As time passes, the minute hand rotates to point at different numbers (the terminal side). If it points at 1, that's a certain angle, and if after a full rotation it points at 1 again, we have created two different angles that are coterminal, as their initial and terminal sides are the same.

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Most popular questions from this chapter

Graph the functions \(f\) and \(g\). Use the graphs to make a conjecture about the relationship between the functions. $$ f(x)=\sin ^{2} x, \quad g(x)=\frac{1}{2}(1-\cos 2 x) $$

Use a graphing utility to graph the function. Include two full periods. $$ y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right) $$

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Evaluate the expression without using a calculator. $$ \arcsin 0 $$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$ g(x)=e^{-x^{2} / 2} \sin x $$
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