I was doing a trig question asking to find the reference angle of -500 degrees. I got the right number (40), but when I checked with the answer key it said that it was positive, which got me thinking: why is the reference angle in this situation positive if the original angle is negative? And if that's the case, why isn't the reference angle of 500 degrees -40?
Why is the reference angle always positive?
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Steven G
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The flaw with your argument that the reference should be negative is because the location of -500 is the same as -500 + 360 + 360 = +220. Now the reference angle for -500 and +220 are the same.
When you draw the triangle, in quadrant 3 for this particular angle, the angles are all positive as ALL angles in a triangle are positive.
When you draw the triangle, in quadrant 3 for this particular angle, the angles are all positive as ALL angles in a triangle are positive.
The Highlander
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By definition: the Reference Angle is the acute angle between the terminal arm/terminal side of the angle concerned and the x-axis. The reference angle is always positive!
In other words, the reference angle is the angle 'sandwiched' between any angle and the x-axis. It must be less than (or equal to) 90 degrees, and always positive. (Even if it "lies" in the 3rd or 4th quadrant).
Dr.Peterson
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What definition were you given for "reference angle"?
The way I understand it, it is defined as the positive acute angle between the terminal ray and the x-axis. See, for example, here.
That's just the definition; no reason is needed! But one could give good reasons for preferring this ... such as the way the concept is used.