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Chapter 1: Problem 86

Find all angles that are coterminal with the given angle. $$ -60^{\circ} $$

Short Answer

Expert verified
Angles coterminal with \(-60^{\circ}\) can be found using \(-60^{\circ} + 360^{\circ} \cdot k\), where \(k\) is any integer.

Step by step solution

01

Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find coterminal angles, you add or subtract full rotations (360掳) to the given angle.
02

Starting with the Given Angle

We begin with the angle \(-60^{\circ}\). This is the angle in question for which we need to find coterminal angles.
03

Adding a Full Rotation

To find a coterminal angle, add \(360^{\circ}\) to \(-60^{\circ}\):\[-60^{\circ} + 360^{\circ} = 300^{\circ}.\]Thus, \(300^{\circ}\) is a positive coterminal angle.
04

Subtracting a Full Rotation

Next, subtract \(360^{\circ}\) from \(-60^{\circ}\):\[-60^{\circ} - 360^{\circ} = -420^{\circ}.\]Thus, \(-420^{\circ}\) is another coterminal angle.
05

General Formula for Coterminal Angles

To find any coterminal angle, use the formula:\[\theta + 360^{\circ} \cdot k,\]where \(\theta = -60^{\circ}\) and \(k\) is any integer. This formula allows you to find infinitely many coterminal angles.

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Key Concepts

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

Understanding Angle Measurement
Angle measurement is a way to describe the space between two intersecting lines or rays, typically in degrees or radians. It is important to note the sign of the angle as it indicates direction:
  • Positive angles represent counterclockwise rotations from a standard position.
  • Negative angles represent clockwise rotations.

This concept is fundamental when working with angles in mathematics because it determines how you interpret rotations. For instance, a -60掳 angle means a 60掳 rotation in the clockwise direction from the positive x-axis. Therefore, understanding whether the angle is positive or negative helps in determining its placement on the unit circle or coordinate plane.
What are Full Rotations?
A full rotation involves rotating completely around a circle back to the original position. In degree measurement, a full rotation is equivalent to 360掳. This is because a circle has 360 degrees in total.
To find coterminal angles, you often add or subtract these full rotations to an angle. This doesn't change the angle's terminal position, hence making them 'coterminal.' For example:
  • Adding 360掳 to -60掳:
    • -60掳 + 360掳 = 300掳
  • Subtracting 360掳 from -60掳:
    • -60掳 - 360掳 = -420掳
Both 300掳 and -420掳 point to the same terminal side as -60掳 in a coordinate system, showcasing the idea of coterminal angles effectively.
Understanding Standard Position
In mathematics, angles are often discussed in terms of their "standard position." An angle is in standard position when:
  • Its vertex is located at the origin (0, 0) in a two-dimensional plane.
  • The initial side aligns with the positive x-axis.
  • The terminal side is where the angle measurement ends, based on its magnitude and direction.

When you draw an angle in standard position, you can easily visualize how different coterminal angles share the same terminal side while starting from a common point. This is why knowing that angles in standard position all start from the positive x-axis helps in conceptualizing how adding or subtracting full 360掳 rotations will land you back on the same line. This concept ties together with coterminal angles as it clearly depicts their cyclical nature.

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Most popular questions from this chapter

Find \(\tan \theta\) if \(\theta\) is the angle formed by the line \(y=m x\) and the positive \(x\)-axis (Figure 2).

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . If \(\sin \theta=-\frac{5}{13}\) and \(\theta\) terminates in QIV, find \(\cos \theta\)

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ (1+\sin \theta)(1-\sin \theta)=\cos ^{2} \theta $$

Simplify the expression \(\sqrt{64-4 x^{2}}\) as much as possible after substituting \(4 \sin \theta\) for \(x\).

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\cos \theta\) if \(\sin \theta=-\frac{1}{4}\) and \(\theta\) terminates in QII.
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