Instruction

1

If you need to calculate the angle

*of inclination*of a straight line to the x-axis, and you don't know the equation of the line, omit from any point of the line (except the point of intersection with the axis) perpendicular to the axis. Then measure the other two sides received a right triangle and find the ratio of adjacent side to the opposite. The resulting number will be equal to the tangent**of the angle***of inclination*. This method is useful not only to explore**the angle***of inclination of*a straight line, but also to measure any angles, as in the drawing, and in life (for example, the angle of the slope of the roof).2

If you know the equation of the line and you need to find the tangent

**of the angle***of inclination*of this straight line to the x-axis, Express through H. as a result, you will get an expression of type y=KX+b. Note the k-factor is the tangent**of the angle***of inclination*between the positive direction of x-axis and the beam direct located need this axis. If k=0, the tangent is also zero, then there is a direct parallel to or coincides with the abscissa axis.3

If you are given a complicated function such as a quadratic, and you need to find the tangent

**of the angle***of inclination of the*tangent to this function, or, in other words, the angle coefficient, and calculate the derivative. Then calculate the value of the derivative at a given point, which will be held tangent. The resulting number is**the tangent****of the angle***of inclination of the*tangent. For example, you are given the function y=x^2+3x, the derivative, you will get an expression y`=2x+3. To find the slope at x=3, substitute this value into the equation. As a result of simple calculations it is easy to obtain y=2*3+3=9, this is the desired tangent.4

To find the tangent

**of the angle***of inclination*of one side of the triangle to another, proceed as follows. Find the sine (sin) of that**angle**and divide it by the cosine (cos), you get the tangent of that**angle**.# Advice 2 : How to find the tangent if we know the cosine

The concept of a

**tangent**is one of the basic in trigonometry. It represents some trigonometric function which is periodic, but not continuous in the area of definitions of sine and**cosine**. And has discontinuities at the points (+,-)PI*n+PI/2, where n is the period of the function. In Russia he is referred to as tg(x). It can be represented using any trigonometric function, as they are all closely interrelated.You will need

- Textbook on trigonometry.

Instruction

1

In order to Express the tangent of an angle using a sine, you need to recall a geometric definition

**of tangent**. So, the tangent of an acute angle in a right triangle is the ratio opposite over adjacent.2

On the other hand, consider a Cartesian coordinate system where the unit circle is drawn with radius R=1 and with center in the origin. Take counterclockwise rotation as positive and the opposite direction negative.

3

Mark a certain point M on the circle. It will drop a perpendicular on the axis Oh, call it point N. get the triangle OMN, in which the angle ONM is a direct.

4

Now consider acute angle MON, by definition of sine and

sin(MON) = MN/OM cos(MON) = ON/OM. Then MN= sin(MON)*OM and ON = cos(MON)*OM.

**cosine**and an acute angle in a right trianglesin(MON) = MN/OM cos(MON) = ON/OM. Then MN= sin(MON)*OM and ON = cos(MON)*OM.

5

Returning to the geometric definition

tg(MON) = sin(MON)*OM/cos(MON)*OM cut OM, then tg(MON) = sin(MON)/cos(MON).

**of the tangent**(tg(MON) = MN/ON), substitute the above expression. Then:tg(MON) = sin(MON)*OM/cos(MON)*OM cut OM, then tg(MON) = sin(MON)/cos(MON).

6

From the basic trigonometric identities (sin^2(x)+cos^2(x)=1) Express

**the cosine**, using the sine: cos(x)=(1-sin^2(x))^0,5 Substitute the expression obtained in step 5. Then tg(MON) = sin(MON)/(1-sin^2(MON))^0,5.7

Sometimes there is a need to calculate the

= 2*sin(x)/(1-sin^2(x))^0,5/(1-sin^2(x)/(1-sin^2(x)).

**tangent**double-angle and half-hearted. Here, too, relations are derived:tg(x/2) = (1-cos(x))/sin(x) = (1-(1-sin^2(x))^0,5)/sin(x);tg(2x) = 2*tg(x)/(1-tg^2(x)) = 2*sin(x)/(1-sin^2(x))^0,5/(1-sin(x)/(1-sin^2(x))^0,5)^2) == 2*sin(x)/(1-sin^2(x))^0,5/(1-sin^2(x)/(1-sin^2(x)).

8

It is also possible to Express the square

**of the tangent of**a double angle**cosine**and or sine. tg^2(x) = (1-cos(2x))/(1+cos(2x)) = (1-1+2*sin^2(x))/(1+1-2*sin^2(x)) = (sin^2(x))/(1-sin^2(x)).Note

Pay attention to the tolerance range when the solution of equations and inequalities.

Useful advice

Knowledge by heart of the main identities, which will help you to quickly move from one trigonometric function to another.

# Advice 3 : How to find the slope of the straight

The angular coefficient of the straight line — coefficient k in the equation y = kx + b of a straight line on the coordinate plane is numerically equal to the tangent of the angle (which is the smallest rotation from axis Ox to the axis Oy) between the positive direction of x-axis and the straight line.

You will need

- Math.

Instruction

1

Write the equation of a straight line and Express the ordinate of a function through the x point. For example, let the equation of the line: 3x + 4y = 13. Express the y coordinate: y = -3x/4 + 13/4.

2

The coefficient in front of x will be the slope of a straight line in the Cartesian coordinate system. That is, the slope of k=-3/4.

3

To find the angle between line and x-axis, it is sufficient to calculate the arc tangent of the angular coefficient. Thus the angle between line 3x + 4y = 13 and x-axis is equal to: U = artg(-3/4) = -36 degrees.

Note

Since the coefficient is equal to the tangent of the angle of inclination, the angle varies from -90 degrees to +90 degrees.

Useful advice

Knowing the coordinates of a direction vector of the straight, you can always find the angle between it and the x-axis, and hence the slope of the straight line.

# Advice 4 : How to calculate the angle of the triangle

Tre

**gon**define its corners and sides. The type of corners of such a triangle**a square**and a sharp – all three angles acute obtuse one angle obtuse, right angle – one angle of a straight line, an equilateral triangle,**a square**, e all the angles equal 60. To find the angle, friction**angle**and in different ways depending on the source data.You will need

- basic knowledge of trigonometry and geometry

Instruction

1

Calculate the angle, friction

**angle**and, if known to the other two angle α and β, as the difference between 180°−(α+β) as the sum of the angles in**a square**, e is always equal to 180°. For example, suppose there are two angle friction**angle**and α=64°, β=45°, then the unknown angle γ=180−(64+45)=71°.2

Use the theorem of cosines when you know the lengths of two sides a and b friction

**angle**a and angle α between them. Find the third side by the formula c=√(a2+b2−2*a*b*cos(α)), as the squared length of any side of triangle**a square**and is equal to the sum of the squares of the lengths of the other sides minus twice the product of the lengths of these sides into the cosine of the angle between them. Write down the theorem of cosines for the other two sides: a2=b2+c2−2*b*c*cos(β), b2=a2+c2−2*a*c*cos(γ). Express of these formulas, the unknown angles: β=arccos((b2+c2−a2)/(2*b*c)), γ=arccos((a2+c2−b2)/(2*a*c)). For example, suppose tre**gon**e well-known side a=59, b=27, the angle between them α=47°. Then the unknown side c=√(592+272-2*59*27*cos(47°))≈45. Then β=arccos((272+452-592)/(2*27*45))≈107°, γ=arccos((592+452-272)/(2*59*45))≈26°.3

Find the angles of triangle

**a square**and, if you know the lengths of all three sides a, b and c tre**gon**. To do this, calculate the area of triangle**angle**and by Heron's formula: S=√(p*(p−a)*(p−b)*(p−c)), where p=(a+b+c)/2 – properiter. On the other hand, since the area of triangle**angle**and is equal to S=0,5*a*b*sin(α), we Express from this formula the angle α=arcsin(2*S/(a*b)). Similarly, β=arcsin(2*S/(b*c)), γ=arcsin(2*S/(a*c)). For example, suppose that we are given tre**a square**with sides a=25, b=23 and C=32. Then count properiter p=(25+23+32)/2=40. Calculate the area by Heron's formula: S=√(40*(40-25)*(40-23)*(40-32))=√(40*15*17*8)=√(81600)≈286. Find the angles: α=arcsin(2*286/(25*23))≈84°, β=arcsin(2*286/(23*32))≈51°, and the angle γ=180−(84+51)=45°.# Advice 5 : How to determine the angles of inclination of the plane

In the production of various works on the country or the plot (laying various sites, paving slabs or paths) often have to stitch the oblique tracks site, located on different levels. It is necessary to carefully define and maintain the inclination angles of the plane on such sites.

You will need

- - vertical or horizontal building level;
- - plumb
- - protractor;
- - protractor;
- - smooth wooden beam of length 1.5 m;
- laser level measuring ruler;
- - hydrolevel any marker, peg 2;
- roulette.

Instruction

1

To determine the angle of inclination of the plane the simplest way to use a plumb line, a wooden block and protractor. Put the timber to check the plane. Left hand hold the plumb line at an altitude of 300 – 400 mm. Move the plumb line to the edge of the timber. Reassure the lower part of the slope. The right hand vertically set the protractor with the flat side on the Board. Move the protractor to align the reference point of the protractor with the string of the plumb. Consider the angle of the plane intersection line of the plumb with the scale of the protractor. Get the angle of the plane relative to the vertical. If the desired angle relative to the horizon, calculate it, subtracting from the received angle of 90. This method is use for rough measurements, as it gives a low measurement precision of the angle of inclination of the plane.

2

More specific the following measurement method. Put the timber to check the plane. On the edge of the beam vertically, check the level. Level keep left hand. The right hand attach the protractor to the sides of the angle. On a scale of protractor think of the angle of inclination of the plane.

3

The most accurate method, which used laser level. Install horizontally the base level. Turn on the laser head. Measure the level of elevation from the horizontal of the laser beam to the surface of the plane check on the section length of 1 m. When the value of the differential up to 1 m in this area every of 2.22 cm drop close to 1 degree.

4

High precision of tilt measurement can be achieved using hydrolevel any is a laser level. Measurement hammer parallel to the tilt plane of the two markers at a distance of 1 m. they Mark the horizon with hidrorema. Measure the distance from the labels to the horizon plane. Subtract the larger from the smaller size – get the size of the difference in height at a distance of meters. This value divide by 2.22 and get the angle of the measured area of the plane in degrees.

# Advice 6 : How to find the slope of the tangent

Y=f(x) will be tangent to the shown in figure the schedule at the point x0 if it passes through the point with coordinates (x0; f(x0)) and has the slope f'(x0). To find such a factor, knowing the characteristics of a tangent, it's easy.

You will need

- mathematical Handbook;
- - pencil;
- - notebook;
- - protractor;
- a pair of compasses;
- - handle.

Instruction

1

Please note that the schedule is differentiable at the point x0 of the function f(x) does not differ from the segment tangent. In view of this, it is rather close to the segment l, which passes through the point (x0, f(x0)) and (x0+Δx f(x0 + Δx)). To set a line that passes through a certain point And the coefficients (x0, f(x0)), specify its angular coefficient. The angular coefficient is equal to Δy/Δx secant tangent (DF→0) and tends to the number f‘(x0).

2

If the values of f‘(x0) exists, then a tangent or not, or it is held vertically. Because of this, the presence of the derivative of the function at the point x0 due to the existence applied on other than vertical tangent touching the graph of the function at the point (x0, f(x0)). In this case, the slope of the tangent is equal to f'(x0). Thus, it becomes clear the geometric meaning of derivative – calculation of the angular coefficient of the tangent.

3

Draw the picture for more tangents that would be in contact with the graph of the function at the points x1, x2 and X3, and also note the angles formed by these tangents with the abscissa axis (this angle is counted positive in the direction from the axis to the tangent line). For example, the first angle, that is, α1, will be sharp, the second (α2) is stupid, and third (α3) is equal to zero, as conducted straight line parallel to OX axis. In this case, the tangent of the obtuse angle is a negative value, the tangent of an acute angle is positive, and tg0 if the result is zero.

Note

Correctly determine the angle formed by a tangent. To do this, use the protractor.

Useful advice

Two straight inclined will be parallel in that case, if their angular coefficients are equal; perpendicular if the product of the angular coefficients of these tangents equal -1.

# Advice 7 : How to find the tangent using the cosine

Cosine, and sine, referred to as "direct" trigonometric functions. The tangent (along with the cotangent) is referred to the other pair, called "derivatives". There are several definitions of these functions that make possible finding the tangent of a specified angle for a known cosine value from the same value.

Instruction

1

Subtract from unity the quotient of the units on the squared value of the cosine of the specified angle, and from the result, extract the square root, this will be the value of the tangent of the angle, expressed via its cosine: tg(α)=√(1-1/(cos(α))2). In this case, note that in the formula for the cosine is in the denominator of the fraction. The impossibility of division by zero eliminates the use of this expression for angles equal to 90°, and differs from this value by multiples of 180° (270°, 450°, -90° etc.).

2

There is an alternative method of calculating the tangent at the known value of the cosine. It can be used, if not set a limit on the use of other trigonometric functions. To implement this method, first determine the angle for a known cosine value - this can be done using the inverse cosine. Then just calculate the tangent for the angle values. In General this algorithm can be written as: tg(α)=tg(arccos(cos(α))).

3

There are even more exotic option using the definition of cosine and tangent of acute angles using right triangle. The cosine in this definition corresponds to the ratio of the length adjacent to the considered corner of the leg to the length of the hypotenuse. Knowing the cosine ratio to find the corresponding lengths of these two sides. For example, if cos(α)=0.5, then the adjacent side can be taken equal to 10 cm and the hypotenuse is 20cm. The specific numbers here do not matter - the same and right decision you will get to any values having the same ratio. Then by the Pythagorean theorem determine the length of the missing side opposite leg. It will be equal to the square root of the difference between the lengths squared of the hypotenuse and the known leg: √(202-102)=√300. The tangent is, by definition, corresponds to the ratio of the lengths opposite and adjacent sides (√300/10) - calculate it and get the tangent value found using the classic definition of cosine.

# Advice 8 : How to find the tangent of an angle tangent

The geometric meaning of the first order derivative of the function F(x) is a tangent line to its graphics, which passes through a given point of the curve and matching it at this point. Moreover, the value of the derivative at a given point x0 is the slope or tangent of the angle of inclination of the tangent line k = tg a = F`(x0). The calculation of this ratio is one of the most common tasks of the theory of functions.

Instruction

1

Write the given function F(x), e.g. F(x) = (x3 + 15x +26). If the task explicitly specify the point through which the tangent is held, for example, its coordinate x0 = -2, we can dispense with plotting functions and additional straight lines on the Cartesian system OXY. Find the derivative of the first order from the given function F`(x). In this example, F`(x) = (3x2 + 15). Substitute the given value of the argument x0 in the derivative function and calculate its value: F`(-2) = (3(-2)2 + 15) = 27. So you found tg a = 27.

2

When considering tasks where it is required to determine the tangent of the angle of inclination of the tangent to the graph of the function at the point of intersection of this graph with the abscissa axis, you need first to find the numerical value of the coordinates of the point of intersection of the function with OH. For clarity, it is best to build a graph of a function on the two-dimensional plane OXY.

3

Specify the coordinate range for x, for example, from -5 to 5 with step 1. Substituting in the function values x, calculate the corresponding ordinates y and put on the coordinate plane plotting points (x, y). Connect the dots smooth line. You will see the completed chart, the intersection of function and the x-axis. The ordinate of the function at a given point is equal to zero. Find the numerical value of the corresponding argument. For this specified function, for example F(x) = (4x2 - 16), Paranaita to zero. Solve the resulting equation with one variable and calculate x: 4x2 - 16 = 0, x2 = 4, x = 2. Thus, according to the condition of the problem, the tangent of the angle of inclination of the tangent to the graph of the function is to find the point with coordinates x0 = 2.

4

Similarly to the previously described method determine the derivative function: F`(x) = 8*x. Then calculate its value at the point with x0 = 2, which corresponds to the intersection point of the original function with OH. Substitute the obtained value into the derivative function and calculate the tangent of an angle tangent: tg a = F`(2) = 16.

5

When finding the slope at the point of intersection of the function with the ordinate axis (Oh) do the same. Only the coordinate of the initial point x0 should be set to zero.

# Advice 9 : How to find the tangent of an angle in a triangle

The tangent of an angle, like other trigonometric functions, expresses the relationship between the sides and angles of a right triangle. The use of trigonometric functions allows us to replace quantities in the degree measurements on the linear parameters.

Instruction

1

In the presence of protractor angle triangle can be measured and the table Bradis to find the tangent ratio. If it is not possible to determine the degree measure of an angle, determine its tangent using the measurements of the linear gradient shape. To do this, the support construction from an arbitrary point on one of the sides of the angle, drop a perpendicular on the other side. Measure the distance between the ends of the perpendiculars on the sides of the angle, record the measurement in the numerator of the fraction. Now measure the distance from the vertex of the given angle to the vertex angle, i.e. to the point on the side of the angle, which was lowered perpendicular. The number you write in the denominator. Based on the results of measurements of the fraction is equal to the tangent of an angle.

2

The tangent of the angle can be determined by calculation as the ratio of he opposite over adjacent. You can also calculate the tangent using the direct trigonometric functions of the angle sine and cosine. The tangent of an angle is the ratio of the sine of that angle to its cosine. Unlike continuous functions of sine and cosine, the tangent is discontinuous and not defined at the angle of 90 degrees. If there is a zero angle of its tangent equals zero. From ratios of a right triangle it is obvious that the 45 degree angle has a tangent equal to one, since the legs of this right triangle are equal.

3

For small values of angle from 0 to 90 degrees its tangent is positive, since the sine and cosine in that interval is positive. Limits of change of the tangent in this area - from zero to infinitely high values at angles close to straight. Negative values of the angle of the tangent changes sign. The graph of the function Y=tg(x) in the interval -90°