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Chapter 1: Problem 21

Determine whether the angles in each given pair are coterminal. $$ 40^{\circ},-320^{\circ} $$

Short Answer

Expert verified
Yes, 40掳 and -320掳 are coterminal.

Step by step solution

01

Define coterminal angles

Coterminal angles are angles that share the same initial and terminal sides but might have different measures. They differ by integer multiples of 360 degrees.
02

Find coterminal angle for 40掳

To find a coterminal angle of 40掳, add or subtract integer multiples of 360掳. Let's add 360掳 until we get a positive angle: 40掳: 40掳 + 360掳 = 400掳 40掳: 40掳 - 360掳 = -320掳
03

Compare the angles

The angle -320掳 is a coterminal angle of 40掳 because adding -320掳 + 360掳 gives 40掳.

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Key Concepts

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

Angle Measurement
Angle measurement is fundamental for understanding concepts like coterminal angles. Angles are measured in degrees, which is a unit that describes the rotation from an initial side to a terminal side. There are 360 degrees in a full circle. This is deliberate, based on historical and geometric reasons that make a circle easy to divide.

When measuring angles, positive angles indicate counterclockwise rotation and negative angles indicate clockwise rotation. For example, a 40掳 angle means the terminal side is rotated 40 degrees counterclockwise from the initial side. In contrast, a -40掳 angle means the terminal side is rotated 40 degrees clockwise.

Consistency in angle measurement is key to identifying coterminal angles. Always ensure that you're aware of the direction in which an angle is measured.
Initial and Terminal Sides
Understanding initial and terminal sides is crucial when working with angles. An angle is formed by two rays (sides) with a common endpoint known as the vertex. The initial side is the starting position where the rotation begins, and the terminal side is where the rotation ends.

For instance, if you're considering a 40掳 angle, the rotation begins at the initial side and ends at the terminal side after rotating counterclockwise 40 degrees. Similarly, for a -320掳 angle, the rotation starts from the same initial side but ends at the terminal side after rotating clockwise 320 degrees.

It's important to visualize these sides to understand how coterminal angles share the same terminal side despite having different measures. Coterminal angles essentially show different 鈥榯rips鈥� along the circle but land you in the same spot.
Integer Multiples of 360 Degrees
Coterminal angles differ by integer multiples of 360 degrees. This is because 360 degrees represent a full rotation. By adding or subtracting 360 degrees (or any multiple of it), you end up in the same position on the circle.

To determine if two angles are coterminal, check if their measures differ by an integer multiple of 360 degrees. For example, to check if 40掳 and -320掳 are coterminal, we can add 360掳 to 40掳:
40掳 + 360掳 = 400掳
Then subtract 360掳 from 40掳:
40掳 - 360掳 = -320掳

Since we reach -320掳 by subtracting 360掳 from 40掳, these angles are coterminal. This principle helps in simplifying problems and understanding angular rotations better.

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Most popular questions from this chapter

Solve each problem. If \(\tan \alpha=1 / 70,\) then what is \(\cot \alpha ?\)

Find the exact value of each expression for the given value of \(\theta\) Do not use a calculator. $$ \sin (2 \theta) \text { if } \theta=\pi / 8 $$

Solve each problem. Motion of a Spring A weight is suspended on a vertical spring as shown in the accompanying figure. The weight is set in motion and its position \(x\) on the vertical number line in the figure is given by the function $$ x=4 \sin (t)+3 \cos (t) $$ where \(t\) is time in seconds. a. Find the initial position of the weight (its position at times $$ t=0) $$ b. Find the exact position of the weight at time \(t=5 \pi / 4\) seconds.

Cooperative Learming In small groups, discuss the difference between radian measure and degree measure of an angle. Exactly how big is a unit circle? Why is a radian a real number? Could we define radian measure using any size circle?

Find the measure in radians of the smallest positive angle that is coterminal with each given angle. For angles given in terms of \(\pi\) express the answer in terms of \(\pi\). Otherwise, round to the nearest hundredth. $$ -\frac{7 \pi}{6} $$
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