Questions arising from the studytrigonometric functions are diverse. Some of them - about which quarter the cosine is positive and negative, in which quarters the sine is positive and negative. Everything is 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 the graph.
What are the cosine values
If we consider a rectangular triangle, then we have the following aspect ratio, which determines it: the cosine of the angle a is the ratio of the adjacent leg BC to the hypotenuse AB (Figure 1): cos a = BC / AB.
Using the same triangle, one can find the sineangle, tangent and cotangent. The sinus is the ratio of the opposite to the angle of the leg of the AU to the hypotenuse AB. The tangent of the 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 tg a = AC / BC. Cotangent, as the inverse of the tangent function, will be as follows: ctg a = BC / AC.
That is, for identical values of the angleit was found that in a right-angled 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, we must consider a triangle in the Cartesian coordinate system, where both positive and negative values are present.
Visually about a quarter, where is
If you place a rectangular triangle in the first quarter (from 0about up to 90about), where the x and y axes have positive values(segments AO and BO lie on the axes where the values have the sign "+"), then the sine that the cosine will also have positive values, and they are assigned a value with the plus sign. But what happens if you move the triangle to the second quarter (from 90about up to 180about)?
We see that along the axis of the cathode AO 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 is negative. But for the sine, nothing has changed, because to determine its sign, an OB side is needed, which in this case remains 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), you need to look at what mark is assigned to a particular leg. For cosine of an angle a the leg 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.
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, 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 therefore the cosine too.
For the sine, the cases are still “negative”, because the OB leg remains below the starting point O.
In order to understand which quarterscosine is positive, negative, etc., you need to remember the ratio for calculating 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", then you automatically understand that the 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 and 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 visible by moving 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, cosine:
one. On an imaginary circle with a radius equal to one (although, in fact, it does not matter what the radius of a circle is, but in textbooks such an example is most often given; this 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.
Using the first method, you can understand whatit is the sign that depends, and we explained it in detail above. Figure 7, built on this data, visualizes the resulting function and its sign as well as possible.