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单词 DeterminingSignsOfTrigonometricFunctions
释义

determining signs of trigonometric functions


There are at least two mnemonic devices for determining the sign a trigonometric functionDlmfMathworldPlanetmath at a given angle. They are “all snow tastes cold” and (the more mathematical version) “all students take calculusMathworldPlanetmath”.

The first in both of these, “all”, indicates that, if an angle lies in the first quadrantMathworldPlanetmath, then, when any trigonometric function is applied to it, the result is positive.

The second in both of these starts with the letter “s”, which indicates that, if the terminal ray of an angle lies in the second quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are sin and its reciprocal csc.

The third in both of these starts with the letter “t”, which indicates that, if the terminal ray of an angle lies in the third quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are tan and its reciprocal cot.

The fourth in both of these starts with the letter “c”, which indicates that, if the terminal ray of an angle lies in the fourth quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are cos and its reciprocal sec.

Because of how these mnemonic devices work, it is clear that they are in of the calculator trigonometric functions.

Below is a picture that illustrates how the mnemonic device “all students take calculus” works:

all arepositivehereI “all”sinand cscare positivehere“students” II“take” IIItanand cotare positivehereIV “calculus”cosand secare positiveherexy..

For angles whose terminal ray lies on the boundary of two quadrants, the matter of determining sign is not as , but it is still possible to do so through use of the mnemonic device. If, for both of the boundary quadrants, the sign the trigonometric function is positive, then the value of the trigonometric function applied to the angle is 1. If, for both of the boundary quadrants, the sign the trigonometric function is negative, then the value of the trigonometric function applied to the angle is -1. If the sign the trigonometric function is different in the two boundary quadrants, then the value of the trigonometric function applied to the angle is either 0 or undefined.

Example: Since the terminal ray of 2π3 lies in the second quadrant, we have that sin(2π3)>0, and cos(2π3)<0.

( In fact, sin(2π3)=32 and cos(2π3)=-12. )

Example: Since the terminal ray of 7π2 lies on the boundary of the third and fourth quadrants and, when csc is applied to any angle whose terminal ray lies in either the third or fourth quadrant, the result is negative, we have that csc(7π2)=-1.

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更新时间:2025/5/4 10:29:41