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Chapter 1: Problem 50

Find the degree measure of the smallest positive angle that is coterminal with each angle. $$ -840^{\circ} $$

Short Answer

Expert verified
240掳

Step by step solution

01

Understand 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 full circles. Since one full circle is 360掳, let's calculate the coterminal angle by adding 360掳 until we get a positive angle.
02

Add 360掳 Until Positive

Start with -840掳 and continue adding 360掳: -840掳 + 360掳 = -480掳 -480掳 is still negative, so add 360掳 again: -480掳 + 360掳 = -120掳 -120掳 is still negative, so add 360掳 once more: -120掳 + 360掳 = 240掳Now we have a positive angle, 240掳.
03

Verify the Smallest Positive Angle

After adding 360掳 three times, the resulting angle is 240掳, which is positive and coterminal with -840掳. Therefore, the smallest positive angle coterminal with -840掳 is 240掳.

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Key Concepts

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

degree measure
Angles can be measured in degrees, which is a way to segment a circle into 360 equal parts.
This is similar to how a clock divides one hour into 60 minutes.
One full circle is equivalent to 360 degrees (掳).
When working with angles, it鈥檚 essential to understand that degrees help describe the angle's size in relation to a circle.
For example, a 90掳 angle is a quarter of a full circle.
A 180掳 angle is a half-circle.
Degrees make it simpler to understand and visualize angles within circles and other geometric shapes.
positive angle
A positive angle goes counterclockwise from the initial side.
This direction is standard when measuring angles.
For instance, moving counterclockwise around a circle starting from the positive x-axis gives positive angle measurements.
Sometimes, you might have a negative angle, meaning it goes clockwise.
To convert a negative angle to a positive one, you can add 360掳 until the angle becomes positive.
This is useful in finding coterminal angles.
In our example, -840掳 is negative, so we keep adding 360掳 until we get a positive angle.
This results in 240掳, our smallest positive coterminal angle.
full circle
A full circle is 360掳, representing one complete rotation around the center point.
When dealing with coterminal angles, understanding the concept of a full circle is important.
By adding or subtracting 360掳, we find angles that are coterminal, or share the same position.
For instance, starting with -840掳 and continuously adding 360掳, we eventually reach a positive angle, 240掳, demonstrating that 240掳 and -840掳 are coterminal.
Remember, knowing that a full circle is 360掳 will help you determine coterminal angles efficiently.

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