Questions arising from the studytrigonometric 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 is familiar with the principle of constructing functions on a graph.
What are the cosine values
If we consider a right triangle, then we have the following aspect ratio, which defines it: with the cosine of an angle a is the ratio of the adjacent BC to the hypotenuse AB (Fig. 1): cos a = Sun / AB.
Using the same triangle you can find the sineangle, tangent and cotangent. The sine is the ratio of the opposite to the angle of the leg AC to the hypotenuse AB. The tangent of the angle is, 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 get that tg a = AC / VS. Cotangent, as a function inverse to the tangent, will be as follows: ctg a = SU / AU.
That is, at the same angle valuesit was found that in a right triangle the aspect ratio is always the same. It would seem, it became clear where these values come from, but why are negative numbers obtained?
To do this, consider the triangle in the Cartesian coordinate system, where there are both positive and negative values.
Clearly about the quarter, where some
First quarter
If you place a right triangle in the first quarter (from 0about up to 90about), where the x and y axes are positive(segments of AO and VO lie on the axes where the values have the "+" sign), then the sine, that cosine will also have positive values, and they are assigned a value with a plus sign. But what happens if you move the triangle to the second quarter (from 90about up to 180about)?
Second quarter
We see that along the axis of the legs of the joint-stock company received a negative value. Cosine of angle a now has this side with a minus ratiotherefore, its final value becomes negative. It turns out that what quarter the cosine is positive depends on the placement 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, the side of the OB is needed, which remained in this case with a plus sign. Let's summarize the first two quarters.
To find out which quarters are cosinespositive, and in which negative (as well as sine and other trigonometric functions), it is necessary to look at what mark is assigned to a particular leg. For cosine of an angle a The arm of the joint-stock company is important, for a sine - OV.
The first quarter has so far become the only one answering the question: “In which quarters is the sine and cosine positive at the same time?”. Let us see further whether there will be more matches on the sign of these two functions.
In the second quarter of the leg, the joint-stock company began to have a negative value, which means that the cosine became negative. For sine, a positive value is stored.
Third quarter
Now both legs AO and OB became negative. Recall the relations for cosine and sine:
Cos a = AO / AB;
Sin a = BO / AB.
AB always has a positive sign in thiscoordinate 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 with numbers, among which one and only one has a minus sign, then the result will also be with this sign.
The result at this stage:
1) What quarter is the cosine positive? In the first of three.
2) In which quarter is the sine positive? In the first and second of the three.
Fourth quarter (from 270about up to 360about)
Here, the leg AO re-acquires the plus sign, and hence the cosine too.
For the sine, the cases are still “negative”, because the OB leg remains below the starting point O.
conclusions
In order to understand which quarterscosine is positive, negative, etc., you need to remember the ratio to calculate the cosine: adjacent to the angle of the leg, divided by the hypotenuse. Some teachers suggest memorizing this: to (osinus) = (k) corner. If you remember this "cheat", you automatically understand that a sine is the ratio of the opposite to the angle of the leg to the hypotenuse.
Remember which quarters are cosinespositive, and in which negative, quite difficult. There are many trigonometric functions, and they all have their own values. But still, as a result: positive values for sine are 1, 2 quarters (from 0about up to 180about); for cosine of 1, 4 quarters (from 0about up to 90about from 270about up to 360about). In the remaining quarters, functions have minus values.
It may be easier for someone to remember where the sign is, in the function image.
For sine, it can be seen that from zero to 180about the ridge is above the sin (x) value line,so the function is positive here. For cosine it is the same: in which quarter the cosine is positive (photo 7), and in which quarter it is seen by the displacement of the line above and below the axis cos (x). As a result, we can remember two ways of determining the sign of the functions sine and cosine:
one.On an imaginary circle with a radius equal to one (although, in fact, it does not matter what the radius of the circle is, but in textbooks such an example is most often given; it eases the perception, but at the same time, if you don’t make a reservation that this is not important, children may get confused).
2. In the image, the dependence of the function in (x) on the argument x itself, as in the last figure.
With the first method, you can understand whatit is the sign that depends, and we explained it in detail above. Figure 7, constructed from this data, visualizes the obtained function and its sign as well as possible.
