The topic “The angular coefficient of a tangent as the tangent of the angle of inclination” is given several tasks in the certification exam. Depending on their condition, the graduate may be required to provide either a full answer or a short answer. When preparing to take the Unified State Examination in mathematics, the student should definitely repeat the tasks that require calculating the slope of a tangent.
The Shkolkovo educational portal will help you do this. Our specialists prepared and presented theoretical and practical material in the most accessible way possible. Having become familiar with it, graduates with any level of training will be able to successfully solve problems related to derivatives in which it is necessary to find the tangent of the tangent angle.
Basic moments
To find the correct and rational solution to such tasks in the Unified State Exam, it is necessary to remember the basic definition: the derivative represents the rate of change of a function; it is equal to the tangent of the tangent angle drawn to the graph of the function at a certain point. It is equally important to complete the drawing. It will allow you to find the correct solution to USE problems on the derivative, in which you need to calculate the tangent of the tangent angle. For clarity, it is best to plot the graph on the OXY plane.
If you have already familiarized yourself with the basic material on the topic of derivatives and are ready to start solving problems on calculating the tangent of the tangent angle, similar to the Unified State Examination tasks, you can do this online. For each task, for example, problems on the topic “Relationship of a derivative with the speed and acceleration of a body,” we wrote down the correct answer and solution algorithm. At the same time, students can practice performing tasks of varying levels of complexity. If necessary, the exercise can be saved in the “Favorites” section so that you can discuss the solution with the teacher later.
Numerically equal to the tangent of the angle (constituting the smallest rotation from the Ox axis to the Oy axis) between the positive direction of the abscissa axis and the given straight line.
The tangent of an angle can be calculated as the ratio of the opposite side to the adjacent side. k is always equal to , that is, the derivative of the equation of a straight line with respect to x.
For positive values of the slope k and zero shift coefficient b the straight line will lie in the first and third quadrants (in which x And y both positive and negative). At the same time, large values of the angular coefficient k a steeper straight line will correspond, and a flatter one will correspond to smaller ones.
Straight and perpendicular if , and parallel if .
Notes
Wikimedia Foundation. 2010.
See what “Angular coefficient of a straight line” is in other dictionaries:
slope (direct)- - Topics oil and gas industry EN slope... Technical Translator's Guide
- (mathematical) number k in the equation of a straight line on the plane y = kx+b (see Analytical geometry), characterizing the slope of the straight line relative to the x-axis. In the rectangular coordinate system of U.K. k = tan φ, where φ is the angle between ... ... Great Soviet Encyclopedia
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This term has other meanings, see Direct (meanings). The straight line is one of the basic concepts of geometry, that is, it does not have an exact universal definition. In a systematic presentation of geometry, a straight line is usually taken as one... ... Wikipedia
Image of straight lines in a rectangular coordinate system Straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly defined... ... Wikipedia
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Not to be confused with the term "Ellipsis". Ellipse and its foci Ellipse (ancient Greek ἔλλειψις deficiency, in the sense of lack of eccentricity up to 1) the locus of points M of the Euclidean plane for which the sum of the distances from two given points is F1... ... Wikipedia
Let on a plane where there is a rectangular Cartesian coordinate system, a straight line l passes through point M 0 parallel to the direction vector A (Fig. 96).
If straight l crosses the O axis X(at point N), then at an angle of a straight line l with O axis X we will understand the angle α by which it is necessary to rotate the O axis X around point N in the direction opposite to clockwise rotation, so that the O axis X coincided with a straight line l. (This refers to an angle less than 180°.)
This angle is called inclination angle straight. If straight l parallel to the O axis X, then the angle of inclination is assumed to be zero (Fig. 97).
The tangent of the angle of inclination of a straight line is called slope of a straight line and is usually denoted by the letter k:
tan α = k. (1)
If α = 0, then k= 0; this means that the line is parallel to the O axis X and its slope is zero.
If α = 90°, then k= tan α does not make sense: this means that a straight line perpendicular to the O axis X(i.e. parallel to the O axis at), has no slope.
The slope of a line can be calculated if the coordinates of any two points on this line are known. Let two points on a line be given: M 1 ( x 1 ; at 1) and M 2 ( x 2 ; at 2) and let, for example, 0< α < 90°, а x 2 > x 1 , at 2 > at 1 (Fig. 98).
Then from the right triangle M 1 PM 2 we find
$$ k=tga = \frac(|M_2 P|)(|M_1 P|) = \frac(y_2 - y_1)(x_2 - x_1) $$
$$ k=\frac(y_2 - y_1)(x_2 - x_1) \;\; (2)$$
It is similarly proven that formula (2) is also true in the case of 90°< α < 180°.
Formula (2) becomes meaningless if x 2 - x 1 = 0, i.e. if straight l parallel to the O axis at. There is no slope coefficient for such straight lines.
Task 1. Determine the angular coefficient of the prim passing through the points
M 1 (3; -5) and M 2 (5; -7).
Substituting the coordinates of points M 1 and M 2 into formula (2), we obtain
\(k=\frac(-7-(-5))(5-3)\) or k = -1
Task 2. Determine the slope of the straight line passing through the points M 1 (3; 5) and M 2 (3; -2).
Because x 2 - x 1 = 0, then equality (2) loses its meaning. There is no slope for this straight line. The straight line M 1 M 2 is parallel to the O axis at.
Task 3. Determine the slope of the line passing through the origin and point M 1 (3; -5)
In this case, point M 2 coincides with the origin. Applying formula (2), we obtain
$$ k=\frac(y_2 - y_1)(x_2 - x_1)=\frac(0-(-5))(0-3)= -\frac(5)(3); \;\; k= -\frac(5)(3) $$
Let's create an equation of a straight line with an angle coefficient k, passing through the point
M 1 ( x 1 ; at 1). According to formula (2), the angular coefficient of a straight line is found from the coordinates of its two points. In our case, point M 1 is given, and as the second point we can take any point M( X; at) the desired straight line.
If point M lies on a straight line that passes through point M 1 and has an angular coefficient k, then by virtue of formula (2) we have
$$ \frac(y-y_1)(x-x_1)=k \;\; (3) $$
If point M does not lie on a line, then equality (3) does not hold. Consequently, equality (3) is the equation of the line passing through the point M 1 ( x 1 ; at 1) with slope k; this equation is usually written as
y- y 1 = k(x - x 1). (4)
If the straight line intersects the O axis at at some point (0; b), then equation (4) takes the form
at - b = k (X- 0),
y = kx + b. (5)
This equation is called equation of a straight line with slope k and initial ordinate b.
Task 4. Find the angle of inclination of the straight line √3 x + 3at - 7 = 0.
Let us reduce this equation to the form
$$ y= =\frac(1)(\sqrt3)x + \frac(7)(3) $$
Hence, k= tan α = - 1 / √ 3, from where α = 150°
Task 5. Write an equation for a straight line passing through point P(3; -4) with an angular coefficient k = 2 / 5
Substituting k = 2 / 5 , x 1 = 3, y 1 = - 4 into equation (4), we get
at - (- 4) = 2 / 5 (X- 3) or 2 X - 5at - 26 = 0.
Task 6. Write an equation for a straight line passing through point Q (-3; 4) and a component with the positive direction of the O axis X angle 30°.
If α = 30°, then k= tan 30° = √ 3 / 3 . Substituting into equation (4) the values x 1 , y 1 and k, we get
at -4 = √ 3 / 3 (x+ 3) or √3 x-3y + 12 + 3√3 = 0.
Numerically equal to the tangent of the angle (constituting the smallest rotation from the Ox axis to the Oy axis) between the positive direction of the abscissa axis and the given straight line.
The tangent of an angle can be calculated as the ratio of the opposite side to the adjacent side. k is always equal to , that is, the derivative of the equation of a straight line with respect to x.
For positive values of the slope k and zero shift coefficient b the straight line will lie in the first and third quadrants (in which x And y both positive and negative). At the same time, large values of the angular coefficient k a steeper straight line will correspond, and a flatter one will correspond to smaller ones.
Straight and perpendicular if , and parallel if .
Notes
Wikimedia Foundation. 2010.
- Iphit (king of Elis)
- List of Decrees of the President of the Russian Federation “On awarding state awards” for 2001
See what “Angular coefficient of a straight line” is in other dictionaries:
slope (direct)- - Topics oil and gas industry EN slope... Technical Translator's Guide
Slope factor- (mathematical) number k in the equation of a straight line on the plane y = kx+b (see Analytical geometry), characterizing the slope of the straight line relative to the x-axis. In the rectangular coordinate system of U.K. k = tan φ, where φ is the angle between ... ... Great Soviet Encyclopedia
Equations of a line
ANALYTIC GEOMETRY- a section of geometry that studies the simplest geometric objects using elementary algebra based on the coordinate method. The creation of analytical geometry is usually attributed to R. Descartes, who outlined its foundations in the last chapter of his... ... Collier's Encyclopedia
Reaction time- Reaction time (RT) measurement is probably the most venerable subject in empirical psychology. It originated in the field of astronomy, in 1823, with the measurement of individual differences in the speed of perception of a star crossing a telescope line. These … Psychological Encyclopedia
MATHEMATICAL ANALYSIS- a branch of mathematics that provides methods for quantitative research of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures limited by curved contours and ... Collier's Encyclopedia
Straight- This term has other meanings, see Direct (meanings). The straight line is one of the basic concepts of geometry, that is, it does not have an exact universal definition. In a systematic presentation of geometry, a straight line is usually taken as one... ... Wikipedia
Straight line- Image of straight lines in a rectangular coordinate system Straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly defined... ... Wikipedia
Direct- Image of straight lines in a rectangular coordinate system Straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly defined... ... Wikipedia
Minor shaft- Not to be confused with the term "Ellipsis". Ellipse and its foci Ellipse (ancient Greek ἔλλειψις deficiency, in the sense of lack of eccentricity up to 1) the locus of points M of the Euclidean plane for which the sum of the distances from two given points is F1... ... Wikipedia
In mathematics, one of the parameters that describes the position of a line on the Cartesian coordinate plane is the angular coefficient of this line. This parameter characterizes the slope of the straight line to the abscissa axis. To understand how to find the slope, first recall the general form of the equation of a straight line in the XY coordinate system.
In general, any line can be represented by the expression ax+by=c, where a, b and c are arbitrary real numbers, but a 2 + b 2 ≠ 0.
Using simple transformations, such an equation can be brought to the form y=kx+d, in which k and d are real numbers. The number k is the slope, and the equation of a line of this type is called an equation with a slope. It turns out that to find the slope, you simply need to reduce the original equation to the form indicated above. For a more complete understanding, consider a specific example:
Problem: Find the slope of the line given by the equation 36x - 18y = 108
Solution: Let's transform the original equation.
Answer: The required slope of this line is 2.
If, during the transformation of the equation, we received an expression like x = const and as a result we cannot represent y as a function of x, then we are dealing with a straight line parallel to the X axis. The angular coefficient of such a straight line is equal to infinity.
For lines expressed by an equation like y = const, the slope is zero. This is typical for straight lines parallel to the abscissa axis. For example:
Problem: Find the slope of the line given by the equation 24x + 12y - 4(3y + 7) = 4
Solution: Let's bring the original equation to its general form
24x + 12y - 12y + 28 = 4
It is impossible to express y from the resulting expression, therefore the angular coefficient of this line is equal to infinity, and the line itself will be parallel to the Y axis.
Geometric meaning
For a better understanding, let's look at the picture:
In the figure we see a graph of a function like y = kx. To simplify, let’s take the coefficient c = 0. In the triangle OAB, the ratio of side BA to AO will be equal to the angular coefficient k. At the same time, the ratio BA/AO is the tangent of the acute angle α in the right triangle OAB. It turns out that the angular coefficient of the straight line is equal to the tangent of the angle that this straight line makes with the abscissa axis of the coordinate grid.
Solving the problem of how to find the angular coefficient of a straight line, we find the tangent of the angle between it and the X axis of the coordinate grid. Boundary cases, when the line in question is parallel to the coordinate axes, confirm the above. Indeed, for a straight line described by the equation y=const, the angle between it and the abscissa axis is zero. The tangent of the zero angle is also zero and the slope is also zero.
For straight lines perpendicular to the x-axis and described by the equation x=const, the angle between them and the X-axis is 90 degrees. The tangent of a right angle is equal to infinity, and the angular coefficient of similar straight lines is also equal to infinity, which confirms what was written above.
Tangent slope
A common task often encountered in practice is also to find the slope of a tangent to the graph of a function at a certain point. A tangent is a straight line, therefore the concept of slope is also applicable to it.
To figure out how to find the slope of a tangent, we will need to recall the concept of derivative. The derivative of any function at a certain point is a constant numerically equal to the tangent of the angle that is formed between the tangent at the specified point to the graph of this function and the abscissa axis. It turns out that to determine the angular coefficient of the tangent at the point x 0, we need to calculate the value of the derivative of the original function at this point k = f"(x 0). Let's look at the example:
Problem: Find the slope of the line tangent to the function y = 12x 2 + 2xe x at x = 0.1.
Solution: Find the derivative of the original function in general form
y"(0.1) = 24. 0.1 + 2. 0.1. e 0.1 + 2. e 0.1
Answer: The required slope at point x = 0.1 is 4.831
