In life, we often have to face math problems: at school, at university, and then helping our child with homework. People of certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article, we will analyze one of them: finding the leg of a right triangle.
What is a right triangle
First, let's remember what a right triangle is. A right triangle is a geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides forming a right angle are called the legs, and the side that lies opposite the right angle is called the hypotenuse.
Finding the leg of a right triangle
There are several ways to find out the length of the leg. I would like to consider them in more detail.
Pythagorean theorem to find the leg of a right triangle
If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².
Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).
Trigonometric relations to find the leg of a right triangle
It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using the sine
The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.
Example. The hypotenuse is 10 cm and angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).
Find the leg of a right triangle using cosine
The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos \u003d b / c, where b is the leg adjacent to the given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.
Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).
Find the leg of a right triangle using the tangent
The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.
Example. Angle A is 45 degrees, the hypotenuse is 10 cm. According to the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).
Find the leg of a right triangle using the cotangent
The cotangent of an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "inverted tangent". We get: b=ctg*a.
Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).
So, now you know how to find the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.
We begin our study of trigonometry with a right triangle. Let's define what the sine and cosine are, as well as the tangent and cotangent of an acute angle. These are the basics of trigonometry.
Recall that right angle is an angle equal to . In other words, half of the unfolded corner.
Sharp corner- smaller .
Obtuse angle- larger . In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle is denoted.
An angle is denoted by the corresponding Greek letter.
Hypotenuse A right triangle is the side opposite the right angle.
Legs- sides opposite sharp corners.
The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):
Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.
Let's prove some of them.
1. The sum of the angles of any triangle is . Means, the sum of two acute angles of a right triangle is .
2. On the one hand, as the ratio of the opposite leg to the hypotenuse. On the other hand, since for the angle the leg will be adjacent.
We get that . In other words, .
3. Take the Pythagorean theorem: . Let's divide both parts by:
We got basic trigonometric identity:
Thus, knowing the sine of an angle, we can find its cosine, and vice versa.
4. Dividing both parts of the main trigonometric identity by , we get:
This means that if we are given the tangent of an acute angle, then we can immediately find its cosine.
Likewise,
Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?
This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of sine, cosine, tangent and cotangent values for "good" angles from to.
Notice the two red dashes in the table. For the corresponding values of the angles, the tangent and cotangent do not exist.
Let's analyze several problems in trigonometry from the Bank of FIPI tasks.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Since , we have: .
2. In a triangle, the angle is , , . Find . , is equal to half of the hypotenuse.
Triangle with angles , and is isosceles. In it, the hypotenuse is times larger than the leg.
One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.
Origins of trigonometry
Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.
Historically, right triangles have been the main object of study in this section of mathematical science. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure under consideration using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy, and even art.
First stage
Initially, people talked about the relationship of angles and sides exclusively on the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this section of mathematics.
The study of trigonometry at school today begins with right-angled triangles, after which the acquired knowledge is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because the earth's surface, and the surface of any other planet, is convex, which means that any surface marking will be "arc-shaped" in three-dimensional space.
Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that, according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.
Definition
Finally, with a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.
The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.
So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.
The simplest formulas
In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.
The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you want to know the value of the angle, not the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the conversion rules and a few basic formulas, you can at any time independently derive the required more complex formulas on a sheet of paper.
Double angle formulas and addition of arguments
Two more formulas that you need to learn are related to the values \u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, the pairwise product of the sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself, taking the angle of alpha equal to the angle of beta.
Finally, note that the double angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of \u200b\u200bthe figure, and the size of each side, etc.
The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product, multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Mistakes due to inattention
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's get acquainted with the most popular of them.
First, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer as an ordinary fraction, unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the task, new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values such as the root of three or two, because they occur in tasks at every step. The same applies to rounding "ugly" numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
Finally
So you are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.
How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of the trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here you will be helped by ordinary school mathematics.
Chapter I. Solution of Right Triangles
§3 (37). Basic ratios and tasks
In trigonometry, problems are considered in which it is required to calculate certain elements of a triangle by a sufficient number of numerical values of its given elements. These tasks are usually referred to as decision triangle.
Let ABC be a right triangle, C a right angle, a and b- legs opposite acute angles A and B, with- hypotenuse (Fig. 3);
then we have:
The cosine of an acute angle is the ratio of the adjacent leg to the hypotenuse:
cos A = b/ c, cos B = a / c (1)
The sine of an acute angle is the ratio of the opposite leg to the hypotenuse:
sin A = a / c, sin B = b/ c (2)
The tangent of an acute angle is the ratio of the opposite leg to the adjacent one:
tan A = a / b, tg B = b/ a (3)
The cotangent of an acute angle is the ratio of the adjacent leg to the opposite one:
ctgA= b/ a, ctg B = a / b (4)
The sum of acute angles is 90°.
Basic problems for right triangles.
Task I. Given the hypotenuse and one of the acute angles, calculate the other elements.
Decision. Let given with and A. Angle B = 90° - A is also known; legs are found from formulas (1) and (2).
a = c sinA, b = c cos A.
Task II . Given a leg and one of the acute angles, calculate the other elements.
Decision. Let given a and A. Angle B = 90° - A is known; from formulas (3) and (2) we find:
b = a tg B (= a ctg A), with = a/sin A
Task III. Given the leg and hypotenuse, calculate the remaining elements.
Decision. Let given a and with(and a< с ). From equalities (2) we find the angle A:
sin A = a / c and A = arc sin a / c ,
and finally the leg b:
b = with cos A (= with sin B).
Task IV. Legs a and b are given to find other elements.
Decision. From equalities (3) we find an acute angle, for example A:
tg A = a / b, A = arctan a / b ,
angle B \u003d 90 ° - A,
hypotenuse: c = a/sin A (= b/sinB; = a/cos B)
Below is an example of solving a right triangle using logarithmic tables*.
* The calculation of the elements of right triangles according to natural tables is known from the geometry course of the VIII class.
When calculating using logarithmic tables, one should write out the corresponding formulas, prologarithm them, substitute numerical data, find the required logarithms of known elements (or their trigonometric functions) from the tables, calculate the logarithms of the desired elements (or their trigonometric functions) and find the required elements from the tables.
Example. Dana leg a= 166.1 and hypotenuse with= 187.3; calculate acute angles, other leg and area.
Decision. We have:
sin A = a / c; lg sin A = lg a-lg c; 
A ≈ 62°30", B ≈ 90° - 62°30" ≈ 27°30".
We calculate the leg b:
b = a tg B ; lg b= log b+ lg tg B ; 
The area of a triangle can be calculated using the formula
S=1/2 ab = 0,5 a 2 tg B; 
For control, we calculate the angle A on a slide rule:
A \u003d arc sin a / c= arc sin 166 / 187 ≈ 62°.
Note. leg b can be calculated by the Pythagorean theorem, using the tables of squares and square roots (Tables III and IV):
b= √187,3 2 - 166,1 2 = √35080 - 27590 ≈ 86,54.
Discrepancy with previously obtained value b= 86.48 is explained by the errors of the tables, which give the approximate values of the functions. The result of 86.54 is more accurate.
As you can see, this circle is built in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).
Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.
What is equal to from a triangle? That's right. In addition, we know that is the radius of the unit circle, and therefore, . Substitute this value into our cosine formula. Here's what happens:
And what is equal to from a triangle? Well, of course, ! Substitute the value of the radius into this formula and get:
So, can you tell me what are the coordinates of a point that belongs to the circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! What coordinate does it correspond to? That's right, coordinate! Thus, the point.
And what then are equal and? That's right, let's use the appropriate definitions of tangent and cotangent and get that, a.
What if the angle is larger? Here, for example, as in this picture:
What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle: an angle (as adjacent to an angle). What is the value of the sine, cosine, tangent and cotangent of an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:
Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations are applicable to any rotations of the radius vector.
It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain size, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.
So, we know that a whole revolution of the radius vector around the circle is or. Is it possible to rotate the radius vector by or by? Well, of course you can! In the first case, therefore, the radius vector will make one complete revolution and stop at position or.
In the second case, that is, the radius vector will make three complete revolutions and stop at position or.
Thus, from the above examples, we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.
The figure below shows an angle. The same image corresponds to the corner, and so on. This list can be continued indefinitely. All these angles can be written with the general formula or (where is any integer)
Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:
Here's a unit circle to help you:
Any difficulties? Then let's figure it out. So we know that:
From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner at corresponds to a point with coordinates, therefore:
Does not exist;
Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of trigonometric functions at the corresponding points. Try it yourself first, then check the answers.
Answers:
Does not exist
Does not exist
Does not exist
Does not exist
Thus, we can make the following table:
There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:
But the values \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, must be remembered:
Do not be afraid, now we will show one of the examples rather simple memorization of the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:
Knowing this, you can restore the values for. The numerator " " will match and the denominator " " will match. Cotangent values are transferred in accordance with the arrows shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember the entire value from the table.
Coordinates of a point on a circle
Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?
Well, of course you can! Let's bring out general formula for finding the coordinates of a point.
Here, for example, we have such a circle:
We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the point by degrees.
As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal to. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point the coordinate.
By the same logic, we find the value of the y coordinate for the point. Thus,
So, in general terms, the coordinates of points are determined by the formulas:
Circle center coordinates,
circle radius,
Angle of rotation of the radius vector.
As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:
Well, let's try these formulas for a taste, practicing finding points on a circle?
1. Find the coordinates of a point on a unit circle obtained by turning a point on.
2. Find the coordinates of a point on a unit circle obtained by rotating a point on.
3. Find the coordinates of a point on a unit circle obtained by turning a point on.
4. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
5. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
Having trouble finding the coordinates of a point on a circle?
Solve these five examples (or understand the solution well) and you will learn how to find them!
1.
It can be seen that. And we know what corresponds to a full turn of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
2. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. We know what corresponds to two complete rotations of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
Sine and cosine are tabular values. We remember their values and get:
Thus, the desired point has coordinates.
3. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. Let's depict the considered example in the figure:
The radius makes angles with the axis equal to and. Knowing that the tabular values of the cosine and sine are equal, and having determined that the cosine here takes a negative value, and the sine is positive, we have:
Similar examples are analyzed in more detail when studying the formulas for reducing trigonometric functions in the topic.
Thus, the desired point has coordinates.
4.
Angle of rotation of the radius vector (by condition)
To determine the corresponding signs of sine and cosine, we construct a unit circle and an angle:
As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:
Let's substitute the obtained values into our formula and find the coordinates:
Thus, the desired point has coordinates.
5. To solve this problem, we use formulas in general form, where
The coordinates of the center of the circle (in our example,
Circle radius (by condition)
Angle of rotation of the radius vector (by condition).
Substitute all the values into the formula and get:
and - table values. We remember and substitute them into the formula:
Thus, the desired point has coordinates.
SUMMARY AND BASIC FORMULA
The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.
The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.
The tangent of an angle is the ratio of the opposite (far) leg to the adjacent (close).
The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far).
