In which quarters is the positive cosine? In which quarters is the sine and cosine positive?

The questions arising in the study of trigonometric 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: cosine of angle a is the ratio of the adjacent leg 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 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 get that tan = 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 angle, it 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?

For this, we need to consider a triangle in the Cartesian coordinate system, where both positive and negative values are present.

Visually about a quarter, where what

What is Cartesian coordinates? If we talk about two-dimensional space, we have two directional 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 straight line there are positive numbers, and in the opposite direction - negative. From this, in the final analysis, it directly depends on which quadrates the cosine is positive, and in which, respectively, the negative cosine.

First quarter

If you place a rectangular 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 the sign "+"), then the sine that the cosine Will 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 (90o to 180o)?

Second Quarter

We see that the yoke of the AO has a negative value along the y axis. The cosine of angle a now has this side in the ratio with the minus, therefore its final value becomes negative. It turns out that the fact, in which 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 we need the side of the OB, which remained in this case with a plus sign. Let's sum up the first two quarters.

To find out which quadrants are positive and negative (as well as 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 AO catet is important, and 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?". Let us see further whether there will still be coincidences in the sign of these two functions.

In the second quarter, the AO catheter began to have a negative value, and hence the cosine became negative. A positive value is stored for the sine.

Third quarter

Now both legs of AO and OB became negative. Recall the relations for the cosine and the sine:

Cos a = AO / AB;

Sin a = BO / AB.

AB always has a positive sign in the given coordinate system, since it is not directed to either of the two sides defined by the axes. But the cathets 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, the result will also be with this sign.

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.

The fourth quarter (from 270 ^o to 360 ^o )

Here, the company's cat is again acquiring the plus sign, and hence the cosine, too.

For the sinus, the cases are still "negative", because the OM cathete remained below the initial point O.

conclusions

In order to understand in which quarters the cosine is positive, negative, etc., it is necessary to remember the relation for calculating the cosine: the casset adjacent to the corner divided by the hypotenuse. Some teachers suggest to remember this: to (aspen) = (k) corner. If you remember this "cheat", you automatically understand that the sine is the ratio of the opposite to the angle of the leg to the hypotenuse.

Remember, in which quarters the positive cosine, and in which the negative, is quite difficult. Trigonometric functions are many, and they all have their own meanings. But still, as a result: positive values for the sine - 1, 2 quarters (from 0 to 180 ^o ); For the cosine of 1, 4 quarters (from 0 ^о to 90 ^о and from 270 ^о to 360 ^о ). In the remaining quarters, the functions have a minus sign.

Perhaps, it will be easier for someone to remember where is a sign, according to the image of the function.

For the sine it is seen that from zero to 180 ^{о the} ridge is above the line of the values of sin (x), hence the function here is positive. For the cosine, also: in which quarter is the cosine positive (photo 7), and to which negative is the displacement of the line above and below the cos (x) axis. As a result, we can remember two ways of determining the sign of the sine, cosine functions:

1. On an imaginary circle with a radius equal to one (although, in fact, it does not matter what radius is in the circle, but in textbooks most often this example is given, this facilitates perception, but at the same time, if not to stipulate that this Not important, children can get confused).

2. From the depiction of the dependence of the function on (x) on the argument x itself, as in the last figure.

With the help of the first method, you can UNDERSTAND what the sign depends on, and we have explained it in detail above. Figure 7, constructed from these data, perfectly visualizes the resulting function and its familiarity.