The questions that arise in the study of trigonometric functions are diverse. Some of them are about in which quarters the cosine is positive and negative, in which quarters the sine is positive and negative. Everything turns out to be simple if you know how to calculate the value of these functions in different angles and are familiar with the principle of constructing functions on a graph.
What are the cosine values
If we consider a right-angled triangle, then we have the following aspect ratio that defines it: the cosine of the angle a is the ratio of the adjacent side of the BC to the hypotenuse AB (Fig. 1): cos a = BC / AB.
Using the same triangle, you can find the sine of the angle, tangent and cotangent. The sinus is the ratio of the opposite to the angle of the side of the AC to the hypotenuse AB. The tangent of an angle is found if the sine of the desired angle is divided by the cosine of the same angle; substituting the corresponding formulas for finding the sine and cosine, we obtain that tan a = AC / BC. Cotangent, as the function inverse to the tangent, will be: ctg a = BC / AC.
That is, with the same values of the angle, it was found that in a right triangle the aspect ratio is always the same. It would seem that it became clear where these values came from, but why do negative numbers come from?
To do this, consider the triangle in the Cartesian coordinate system, where there are both positive and negative values.
Clearly about the quarters, where
What are cartesian coordinates? If we talk about two-dimensional space, we have two directed lines that intersect at the point O - this is the abscissa axis (Ox) and the ordinate axis (Oy). From the point O in the direction of the line are positive numbers, and in the opposite direction, negative numbers. In the end, it directly depends on this in which quarters the cosine is positive, and in which, accordingly, negative.
First quarter
If you place a right triangle in the first quarter (from 0 ° to 90 °), where the x and y axis have positive values (the segments AO and BO lie on the axes where the values have a + sign), then there is a sine, that is a cosine, too will have positive values, and they are assigned a value with a plus sign. But what happens if you move the triangle into the second quarter (from 90 about to 180 about )?
Second quarter
We see that along the y axis, the catheter AO received a negative value. The cosine of the angle a now has this side in relation to the minus, and therefore its final value becomes negative. It turns out that the quarter in which the cosine is positive depends on the location of the triangle in the Cartesian coordinate system. And in this case, the cosine of the angle gets a negative value. But for the sine, nothing has changed, because to determine its sign you need the side of the OM, which remained in this case with a plus sign. To summarize the first two quarters.
To find out in which quarters the cosine is positive and in which it is negative (as well as the sine and other trigonometric functions), it is necessary to look at which sign is assigned to one or another leg. For the cosine of the angle a, the side AO is important, for the sine - OB.
The first quarter has so far become the only one that answers the question: “In which quarters is the sine and cosine positive at the same time?” We will see further whether there will still be coincidences in the sign of these two functions.
In the second quarter, the AO cathetus began to have a negative value, which means that the cosine also became negative. A positive value is stored for the sine.
Third quarter
Now both legs AO and OB have become negative. Recall the relations for cosine and sine:
Cos a = AO / AB;
Sin a = BO / AB.
AB always has a positive sign in this coordinate system, since it is not directed to either of the two sides defined by the axes. But the legs became negative, and therefore the result for both functions is also negative, because if you perform multiplication or division operations with numbers, among which one and only one has a minus sign, then the result will also be familiar with this.
The result at this stage:
1) In which quarter is the cosine positive? In the first of three.
2) In which quarter is the sine positive? In the first and second of three.
Fourth quarter (from 270 about to 360 about )
Here the cathetus AO again acquires the plus sign, and hence the cosine too.
For the sine, things are still “negative,” because the catheter OM remained below the starting point O.
conclusions
In order to understand in which quarters the cosine is positive, negative, etc., you need to remember the ratio for calculating the cosine: the leg adjacent to the corner divided by the hypotenuse. Some teachers suggest remembering this way: k (osine) = (k) angle. If you remember this “cheat”, then you automatically understand that the sine is the ratio of the opposite to the side of the leg to hypotenuse.
Remembering in which quarters the cosine is positive and in which negative is quite difficult. There are many trigonometric functions, and all of them have their own meanings. But nevertheless, as a result: positive values for the sine - 1, 2 quarters (from 0 about to 180 about ); for cosines of 1, 4 quarters (from 0 about to 90 about and from 270 about to 360 about ). In the remaining quarters, the functions have minus values.
Perhaps it will be easier for someone to remember where which sign is, according to the image of the function.
For the sine, it is seen that from 0 to 180 ° the crest is located above the line of values of sin (x), which means that the function is positive here. For cosine it is the same: in which quarter the cosine is positive (photo 7), and in which negative it is visible by the movement of the line above and below the axis cos (x). As a result, we can remember two ways to determine the sign of the functions sine, cosine:
1. In an imaginary circle with a radius equal to one (although, in fact, it doesn’t matter what radius the circle has, the textbooks most often give just such an example; this facilitates perception, but at the same time, if you don’t make a reservation that this it doesn’t matter, children can get confused).
2. According to the image of the dependence of the function in (x) on the argument x itself, as in the last figure.
Using the first method, you can UNDERSTAND what the sign depends on, and we have explained this in detail above. Figure 7, constructed on the basis of these data, visualizes the obtained function and its sign belonging as well as possible.