Why does the ti 83 calculator give a positive angle for a negative cosine ratio (second quadrant) but a negative angle for a negative tan ratio? For example for tan -1 the calculator gives an angle of -45 degrees but why not 135 degrees which also has the tan ratio of -1?
ti 83 trig angles
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Otis
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Hi Samson. The inverse to tangent -- aka arctangent or tan-1(x) -- is a function. Do you remember from the definition of 'function' that it's a relationship that maps an input to one and only one output? So, we cannot input a tangent ratio (like -1) and have multiple values come out (like -45º and 135º). Therefore, the range of the inverse tangent function y=tan-1(x) has been constrained to \(-\frac{\pi}{2} < y < \frac{\pi}{2}\) and it's up to the student to add a multiple of \(\pi\) to the calculator's output (if they want an angle in a specific quadrant).
In case you're not familiar with radian measure of angles (used above), I'll repeat the range using degree measure: The inverse tangent function outputs a value in the range -90º < y < 90º and we add whatever multiple of 180º we need to get a specific angle.
EG: -45º + 180º = 135º
The situation is discussed in this Khan Academy video. If you're still unsure, you can google for more discussion or post your questions here. Cheers
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Thank you Otis. But why is the arc tangent function range limited to -90 < x < 90 and the cosine one isn’t?
Otis
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The short answer is because tan(θ) behaves differently than cos(θ). That is, the shape of the tangent graph is different than the shape of the cosine graph. This is why their domains are constrained to different intervals, when forming the corresponding inverse functions.
Again, I'm not sure what instruction you've received about inverse functions (in general, not just in trigonometry). Perhaps, you've already learned that we can write an inverse function only when the original function is one-to-one -- meaning the original function must be continuously increasing or continuously decreasing. Were we to invert an entire curve that's both increasing and decreasing (by reflecting it across the line y=x), the resulting inverse curve would violate the vertical-line test, so it would not represent a functional relationship. Well, all the trig functions increase and decrease across their domains, so an adjustment must be made to ensure that our inverse is still a function. For the original function (like cosine or tangent), we pick an interval within the domain that represents one period wherein the function is one-to-one, and then we constrain the function to that new domain. That constrained domain becomes the range of the inverse function.
Mathematicians have chosen a specific interval for each trig function, to obtain a useful inverse function. Here is a page that shows how these intervals are chosen, and their relationship to both the original function and its inverse (for each of the six trig functions). I hope that the diagrams and explanations help you to visualize why domains and ranges are constrained the way they are, in trigonometry.
There are many more explanations (written and on video) available on the Internet. If you'd like to explore further, you can find other descriptions using a search engine. Also, the more you practice, the clearer the big picture becomes.
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