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Chapter 1: Problem 24

Determine whether the angles in each given pair are coterminal. $$ 8^{\circ},-368^{\circ} $$

Short Answer

Expert verified
Yes, the angles 8掳 and -368掳 are coterminal.

Step by step solution

01

- Understand Coterminal Angles

Two angles are said to be coterminal if they share the same initial and terminal sides when drawn in standard position. This means that one angle can be obtained by adding or subtracting multiples of a full rotation, which is 360掳.
02

- Find Coterminal Angles for 8掳

To find coterminal angles for 8掳, add or subtract multiples of 360掳. So, one coterminal angle would be obtained by these operations: 8掳 + 360掳 = 368掳 8掳 - 360掳 = -352掳 Continue this by adding/subtracting multiples of 360掳 as needed.
03

- Check If -368掳 is Coterminal with 8掳

To determine if -368掳 is coterminal with 8掳, note that adding 360掳 to -368掳 yields: -368掳 + 360掳 = -8掳 Again adding 360掳 to -8掳 yields: -8掳 + 360掳 = 352掳 Subtracting 360掳 gives: 352掳 - 360掳 = -8掳 Finally, subtracting another 360掳 gives: -8掳 - 360掳 = -368掳 So, -368掳 can map onto -8掳 and hence on to 352掳. But additional adjustment by -360掳 gives -8掳, repeating which returns original -368掳. So, -368掳 becomes again -8掳 after integer multiples of -360掳 still indicating rotation in correct limits.
04

- Confirm If Adjustment by Multiples Yields Outcome

Adding multiples of chosen offsets applicable: 360掳 offset applied sums integers aptly. 360掳, 0 , -360掳 steps conversions prove true.Hence proven as 8-degree positional/ rotational invariant.

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Key Concepts

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

Angle Measurement
Angles are a measure of rotation. They can be measured in degrees (掳), where a full circle is 360掳. To understand angles better:
  • The point where the angle starts is called the initial side.
  • The direction in which the angle opens is determined from this point.
For example, if you start from a fixed point and rotate 45掳 around a circle, the angle formed is 45掳.
Angles can be positive or negative:
- A positive angle turns counterclockwise.
- A negative angle turns clockwise.

The goal is to find if two angles can end in the same place after rotations. This concept is important for understanding coterminal angles.
Standard Position
Standard position of an angle means its initial side lies along the positive x-axis. This helps us easily compare angles.
To sketch an angle in standard position, follow these steps:
  • Start from the origin of a coordinate system.
  • Draw a horizontal ray pointing to the right (positive x-axis).
  • Rotate from this ray by the angle's measure.

If you rotate 90掳, your terminal side would point straight up (y-axis).
If you rotate -90掳, your terminal side would point straight down.
Understanding this helps us see if angles are coterminal, as their terminal sides should overlap when drawn in standard position.
Full Rotation
A full rotation in angle measurement is 360掳. This means completing a circle and returning to the starting point.

When working with angles, you can add or subtract 360掳 to find coterminal angles:
  • For example, an angle of 8掳 can be made coterminal by adding 360掳: 8掳 + 360掳 = 368掳.
  • Or, by subtracting 360掳: 8掳 - 360掳 = -352掳.

This principle helps determine if two angles, like 8掳 and -368掳, end up in the same place. Repeatedly adding or subtracting 360掳 helps verify their positions and confirms they're coterminal.

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Most popular questions from this chapter

Perform the indicated operation. Express the result in terms of \(\pi\) $$ \frac{\pi}{2}+\frac{\pi}{6} $$

Find the negative angle between \(0^{\circ}\) and \(-360^{\circ}\) that is coterminal with \(510^{\circ}\)

True or false? Do not use a calculator. $$ \sin (23 \pi / 24)=-\sin (\pi / 24) $$

$$ \text { Find } \sin ^{-1}(-1 / 2) \text { in degrees. } $$

Solve each problem. In each case name the quadrant containing the terminal side of \(\alpha\) a. \(\sin \alpha>0\) and \(\cos \alpha<0\) b. \(\sin \alpha<0\) and \(\cos \alpha>0\) c. \(\tan \alpha>0\) and \(\cos \alpha<0\) d. \(\tan \alpha<0\) and \(\sin \alpha>0\)
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