High School Trigonometry/Ranges of Inverse Circular Functions

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When graphing the inverse trigonometric functions, we run into a problem. Functions may only have one output for each input. We do not encounter this problem in ordinary trigonometric functions. For example,

sin(θ)=x°, in which any value of theta(θ) will have exactly one output, x degrees. Note that occasional inputs of the sine function will yield the same output. For instance,


Both the sin and cosine will yield the same output after 360 degrees, or 2π in radians, is added to the input. The other four functions (tan, cot, sec, csc) have the same output after an addition of 180° or π.

Keep in mind that this keeps with the definition of a function, in which every input has only one output, even if the output may be the same for different input. This is not true with the inverse trig functions!

arcsin(1)={90, 450, 810,... , 90+ n·360} and so on. Therefore, the general consensus is to simply define the domain(input) of arcsin between -90° to 90°, or -π/2 to π/2. Likewise, tangent is also defined this way. Cosine, on the other hand, has its domain defined between 0 and π.

This can be seen most easily with a calculator.

The arcsin(1)={90+n·360} where n is any integer value. However, the calculator will only display the "accepted" values between -90° and 90°. Likewise,

arccos(1)={n·360}. As before, the calculator will display only the accepted values between 0° and 180°.

This material was adapted from the original CK-12 book that can be found here. This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License