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Chapter 4: Problem 10

Determine one positive and one negative coterminal angle for each angle given. $$173^{\circ}$$

Short Answer

Expert verified
Positive coterminal is 533掳, negative coterminal is -187掳.

Step by step solution

01

Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, you can add or subtract multiples of 360掳 since a complete circle is 360掳. The goal is to find one positive coterminal angle and one negative coterminal angle from the given angle of 173掳.
02

Finding the Positive Coterminal Angle

Since the given angle is 173掳, a positive coterminal angle can be found by adding 360掳 to it. Calculate: 173掳 + 360掳 = 533掳. So, 533掳 is a positive coterminal angle of 173掳.
03

Finding the Negative Coterminal Angle

For a negative coterminal angle, subtract 360掳 from the given angle. Calculate: 173掳 - 360掳 = -187掳. Thus, -187掳 is a negative coterminal angle of 173掳.

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Key Concepts

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

Positive Angles
Positive angles are simply angles that are measured in the counter-clockwise direction from the positive x-axis. Imagine standing in a circle facing right鈥攖hat's zero degrees. If you turn left (counter-clockwise) from that position, you are increasing the angle in a positive direction.
Positive angles are always written without a negative sign. They become useful in various applications, especially when figuring out coterminal angles.
  • Example: An angle of 173掳 is a positive angle as it rotates counter-clockwise from the x-axis.
  • Finding coterminal angles in a positive manner often involves adding full rotations or multiples of 360掳.
Negative Angles
Unlike positive angles, negative angles are measured in the clockwise direction from the positive x-axis. Think of them as a reversed version of positive angles.
You might encounter negative angles when you want a quicker or more direct path to a certain angle, especially when finding coterminal angles. They make sense in real-world contexts like determining shortest paths in circular motion.
  • Example: An angle of -187掳 means you have turned 187掳 clockwise from the positive x-axis.
  • To obtain a negative coterminal angle from a positive one, subtract a full rotation or 360掳.
Angle Subtraction
Angle subtraction is a fundamental tool when calculating negative coterminal angles. By subtracting 360掳 from a given angle, you can find a coterminal angle in the clockwise direction, resulting in a negative angle.
This operation helps to identify identical positions on a unit circle despite the difference in angle naming.
  • Example: Subtracting 360掳 from 173掳 gives you -187掳.
  • This shows how a subtraction gives a new perspective on the same angular position.
  • This is useful in scenarios involving rotations where going clockwise saves time or minimizes movement.
Angle Addition
Angle addition is the process generally used to find positive coterminal angles. By adding 360掳 to an angle, you effectively perform one full counter-clockwise rotation, which brings you to the same angular position but with a larger numerical value.
This is the simplest way to calculate a coterminal angle that remains positive.
  • Example: Adding 360掳 to 173掳 results in 533掳.
  • It highlights how angles can maintain the same position while numerically increasing as you complete full rotations around the circle.
  • This addition ensures the result stays positive and provides another perspective respecting the angle's original direction.

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Most popular questions from this chapter

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\). $$\tan 70^{\circ}=\frac{\tan 30^{\circ}+\tan 40^{\circ}}{1-\left(\tan 30^{\circ}\right)\left(\tan 40^{\circ}\right)}$$

A circular gear rotates clockwise by exactly 3.5 revolutions. By how many degrees does it rotate?

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Solve the given problems. Sketch an appropriate figure, unless the figure is given. Two ladders, each \(6.50 \mathrm{m}\) long are leaning against opposite walls of a level alley, with their feet touching. If they make angles of \(38.0^{\circ}\) and \(68.0^{\circ}\) with respect to the alley floor, how wide is the alley?
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