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Chapter 5: Problem 97

Give an expression that generates all angles coterminal with each angle. Let n represent any integer. $$30^{\circ}$$

Short Answer

Expert verified
30掳 + 360掳n

Step by step solution

01

Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They differ by a full rotation, which is 360 degrees.
02

General Formula for Coterminal Angles

To find all angles coterminal with a given angle 胃, we use the formula 胃 + 360掳nwhere n is any integer.
03

Applying the Formula to 30掳

Substitute 胃 = 30掳 into the general formula: 30掳 + 360掳n
04

Simplifying the Expression

The simplified expression for all angles coterminal with 30掳 is: 30掳 + 360掳n

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Key Concepts

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

angle measurement
Understanding angles is crucial in geometry and trigonometry. An angle is formed by two rays that share a common endpoint, called the vertex.
We measure angles in degrees, a unit of angular measurement that represents \(\frac{1}{360}\)th of a full rotation.
The range of possible angles spans from 0 degrees (a flat line) to 360 degrees (a full circle).
Basic types of angles include:
  • Acute angles (less than 90 degrees)
  • Right angles (90 degrees)
  • Obtuse angles (greater than 90 degrees but less than 180 degrees)
  • Straight angles (180 degrees)
  • Reflex angles (greater than 180 degrees but less than 360 degrees)

When working with coterminal angles, we concentrate on how angles can share the same endpoint and initial side despite their numerical differences.
360 degrees
A full rotation around a circle corresponds to 360 degrees. This concept is integral to understanding coterminal angles.
Coterminal angles share the same initial and terminal sides but differ by full rotations.
This means that adding or subtracting multiples of 360 degrees to an angle will yield angles that appear to overlap. Consider 30 degrees, for example:
  • 30掳 + 360掳 = 390掳, which is coterminal with 30掳
  • 30掳 - 360掳 = -330掳, which also shares the same initial and terminal sides as 30掳

This demonstrates how 360 degrees acts as a full circle that we can add or subtract to find coterminal angles.
trigonometry
Trigonometry deals with the study of angles and their relationships to each other. Understanding coterminal angles plays a crucial role in this field.
In trigonometry, angles are not just in the range of 0 to 360 degrees but can also be represented as positive or negative multiples of 360 degrees.
More precisely, if \(\theta\) is an angle, then all angles coterminal with \(\theta\) can be expressed by the formula:
\[\theta + 360掳n \text{where n is any integer}\]\
Applying this to 30 degrees, we get:
\[30掳 + 360掳n \text{ (where n is any integer)}\]\
Understanding trigonometry and these formulas helps solve real-life problems related to waves, oscillations, and circular motion.

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Most popular questions from this chapter

Work each problem. In these exercises, assume the course of a plane or ship is on the indicated bearing. Two docks are located on an east-west line 2587 ft apart. From dock \(A\), the bearing of a coral reef is \(58^{\circ} 22^{\prime}\). From dock \(B\), the bearing of the coral reef is \(328^{\circ} 22^{\prime} .\) Find the distance from dock \(A\) to the coral reef.

As a consequence of Earth's rotation, celestial objects such as the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction perpendicular to Earth's axis.

Decide whether each statement is possible or impossible for some angle \(\theta\). $$\cos \theta=-0.96$$

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ} .(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ},\) and so on. Decide whether each expression is equal to \(0,1,\) or \(-1\) or is undefined. $$\tan \left[(2 n+1) \cdot 90^{\circ}\right]$$

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable. $$-61^{\circ}$$
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