Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. The development of trigonometry began in the days of ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution to the development of this science.
This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning in the context of geometry is explained and illustrated.
Initially, the definitions of trigonometric functions, whose argument is an angle, were expressed through the ratio of the sides of a right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.
The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.
The cotangent of the angle (c t g α) is the ratio of the adjacent leg to the opposite one.
These definitions are given for an acute angle of a right triangle!
Let's give an illustration.
In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.
The definitions of sine, cosine, tangent, and cotangent make it possible to calculate the values of these functions from the known lengths of the sides of a triangle.
Important to remember!
The range of sine and cosine values: from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of tangent and cotangent values is the entire number line, that is, these functions can take any value.
The definitions given above refer to acute angles. In trigonometry, the concept of the angle of rotation is introduced, the value of which, unlike an acute angle, is not limited by frames from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, one can define the sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine a unit circle centered at the origin of the Cartesian coordinate system.
The starting point A with coordinates (1 , 0) rotates around the center of the unit circle by some angle α and goes to point A 1 . The definition is given through the coordinates of the point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of the point A 1 (x, y). sinα = y
Cosine (cos) of the angle of rotation
The cosine of the angle of rotation α is the abscissa of the point A 1 (x, y). cos α = x
Tangent (tg) of rotation angle
The tangent of the angle of rotation α is the ratio of the ordinate of the point A 1 (x, y) to its abscissa. t g α = y x
Cotangent (ctg) of rotation angle
The cotangent of the angle of rotation α is the ratio of the abscissa of the point A 1 (x, y) to its ordinate. c t g α = x y
Sine and cosine are defined for any angle of rotation. This is logical, because the abscissa and ordinate of the point after the rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after the rotation goes to the point with zero abscissa (0 , 1) and (0 , - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined in cases where the ordinate of the point vanishes.
Important to remember!
Sine and cosine are defined for any angles α.
The tangent is defined for all angles except α = 90° + 180° k , k ∈ Z (α = π 2 + π k , k ∈ Z)
The cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)
When solving practical examples, do not say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that from the context it is already clear what is at stake.
Numbers
What about the definition of the sine, cosine, tangent and cotangent of a number, and not the angle of rotation?
Sine, cosine, tangent, cotangent of a number
Sine, cosine, tangent and cotangent of a number t a number is called, which is respectively equal to the sine, cosine, tangent and cotangent in t radian.
For example, the sine of 10 π is equal to the sine of the rotation angle of 10 π rad.
There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.
Any real number t a point on the unit circle is put in correspondence with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are defined in terms of the coordinates of this point.
The starting point on the circle is point A with coordinates (1 , 0).
positive number t
Negative number t corresponds to the point to which the starting point will move if it moves counterclockwise around the circle and passes the path t .
Now that the connection between the number and the point on the circle has been established, we proceed to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of the number t
Sine of a number t- ordinate of the point of the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latter definitions are consistent with and do not contradict the definition given at the beginning of this section. Point on a circle corresponding to a number t, coincides with the point to which the starting point passes after turning through the angle t radian.
Trigonometric functions of angular and numerical argument
Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° · k , k ∈ Z (α = π 2 + π · k , k ∈ Z) corresponds to a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for α = 180 ° k , k ∈ Z (α = π k , k ∈ Z).
We can say that sin α , cos α , t g α , c t g α are functions of the angle alpha, or functions of the angular argument.
Similarly, one can speak of sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a specific value of the sine or cosine of a number t. All numbers other than π 2 + π · k , k ∈ Z, correspond to the value of the tangent. The cotangent is similarly defined for all numbers except π · k , k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the data at the very beginning of the definitions and the angle alpha, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are in full agreement with the geometric definitions given by the ratios of the sides of a right triangle. Let's show it.
Take a unit circle centered on a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw from the resulting point A 1 (x, y) perpendicular to the x-axis. In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y) . The length of the leg opposite the corner is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of the angle α is equal to the ratio of the opposite leg to the hypotenuse.
sin α \u003d A 1 H O A 1 \u003d y 1 \u003d y
This means that the definition of the sine of an acute angle in a right triangle through the aspect ratio is equivalent to the definition of the sine of the angle of rotation α, with alpha lying in the range from 0 to 90 degrees.
Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.
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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 that form 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.
Knowing one of the legs in a right triangle, you can find the second leg and the hypotenuse using trigonometric relationships - the sine and tangent of a known angle. Since the ratio of the leg opposite the angle to the hypotenuse is equal to the sine of this angle, therefore, in order to find the hypotenuse, the leg must be divided by the sine of the angle. a/c=sinα c=a/sinα
The second leg can be found from the tangent of the known angle, as the ratio of the known leg to the tangent. a/b=tanα b=a/tanα
To calculate the unknown angle in a right triangle, you need to subtract the angle α from 90 degrees. β=90°-α
The perimeter and area of \u200b\u200ba right triangle through the leg and the angle opposite to it can be expressed by substituting the previously obtained expressions for the second leg and hypotenuse into the formulas. P=a+b+c=a+a/tanα +a/sinα =a tanα sinα+a sinα+a tanα S=ab/2=a^2/( 2 tanα)
You can also calculate the height through trigonometric relations, but already in the internal right triangle with side a, which it forms. To do this, you need side a, as the hypotenuse of such a triangle, multiplied by the sine of the angle β or the cosine of α, since according to trigonometric identities they are equivalent. (fig. 79.2) h=a cosα
The median of the hypotenuse is equal to half of the hypotenuse or the known leg a divided by two sines α. To find the medians of the legs, we bring the formulas to the appropriate form for the known side and angles. (fig.79.3) m_с=c/2=a/(2 sinα) m_b=√(2a^2+2c^2-b^2)/2=√(2a^2+2a^2+2b^ 2-b^2)/2=√(4a^2+b^2)/2=√(4a^2+a^2/tan^2α)/2=(a√(4 tan^2 α+1))/(2 tanα) m_a=√(2c^2+2b^2-a^2)/2=√(2a^2+2b^2+2b^2-a^2)/ 2=√(4b^2+a^2)/2=√(4b^2+c^2-b^2)/2=√(3 a^2/tan^2α +a^2/sin ^2α)/2=√((3a^2 sin^2α+a^2 tan^2α)/(tan^2α sin^2α))/2=(a√( 3 sin^2α+tan^2α))/(2 tanα sinα)
Since the bisector of a right angle in a triangle is the product of two sides and the root of two, divided by the sum of these sides, replacing one of the legs with the ratio of the known leg to the tangent, we obtain the following expression. Similarly, by substituting the ratio into the second and third formulas, one can calculate the bisectors of the angles α and β. (fig.79.4) l_с=(a a/tanα √2)/(a+a/tanα)=(a^2 √2)/(a tanα+a)=(a√2)/ (tanα+1) l_a=√(bc(a+b+c)(b+c-a))/(b+c)=√(bc((b+c)^2-a^2))/ (b+c)=√(bc(b^2+2bc+c^2-a^2))/(b+c)=√(bc(b^2+2bc+b^2))/(b +c)=√(bc(2b^2+2bc))/(b+c)=(b√(2c(b+c)))/(b+c)=(a/tanα √(2c (a/tanα +c)))/(a/tanα +c)=(a√(2c(a/tanα +c)))/(a+c tanα) l_b=√ (ac(a+b+c)(a+c-b))/(a+c)=(a√(2c(a+c)))/(a+c)=(a√(2c(a+a /sinα)))/(a+a/sinα)=(a sinα √(2c(a+a/sinα)))/(a sinα+a)
The middle line runs parallel to one of the sides of the triangle, while forming another similar right-angled triangle with the same angles, in which all sides are half the size of the original one. Based on this, the middle lines can be found using the following formulas, knowing only the leg and the angle opposite to it. (fig.79.7) M_a=a/2 M_b=b/2=a/(2 tanα) M_c=c/2=a/(2 sinα)
The radius of the inscribed circle is equal to the difference between the legs and the hypotenuse divided by two, and to find the radius of the circumscribed circle, you need to divide the hypotenuse by two. We replace the second leg and the hypotenuse with the ratios of the leg a to the sine and tangent, respectively. (Fig. 79.5, 79.6) r=(a+b-c)/2=(a+a/tanα -a/sinα)/2=(a tanα sinα+a sinα-a tanα)/(2 tanα sinα) R=c/2=a/2sinα
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 mix them up, 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.
Sinus acute angle α of a right triangle is the ratio opposite catheter to the hypotenuse.
It is denoted as follows: sin α.
Cosine acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is denoted as follows: cos α.
Tangent acute angle α is the ratio of the opposite leg to the adjacent leg.
It is denoted as follows: tg α.
Cotangent acute angle α is the ratio of the adjacent leg to the opposite one.
It is designated as follows: ctg α.
The sine, cosine, tangent and cotangent of an angle depend only on the magnitude of the angle.
Rules:
Basic trigonometric identities in a right triangle:
(α - acute angle opposite the leg b and adjacent to the leg a . Side with - hypotenuse. β - the second acute angle).
b | sin 2 α + cos 2 α = 1 | |
a | 1 | |
b | 1 | |
a | 1 1 | |
sinα |
As the acute angle increasessinα andtg α increase, andcos α decreases.
For any acute angle α:
sin (90° - α) = cos α
cos (90° - α) = sin α
Explanatory example:
Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.
Find the sine of angle A and the cosine of angle B.
Decision .
1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of acute angles is 90º, then angle B \u003d 60º:
B \u003d 90º - 30º \u003d 60º.
2) Calculate sin A. We know that the sine is equal to the ratio of the opposite leg to the hypotenuse. For angle A, the opposite leg is side BC. So:
BC 3 1
sin A = -- = - = -
AB 6 2
3) Now we calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC into AB - that is, perform the same actions as when calculating the sine of angle A:
BC 3 1
cos B = -- = - = -
AB 6 2
The result is:
sin A = cos B = 1/2.
sin 30º = cos 60º = 1/2.
From this it follows that in a right triangle the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° - α) = cos α
cos (90° - α) = sin α
Let's check it out again:
1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º - 60º) = cos 60º.
sin 30º = cos 60º.
2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° - 30º) = sin 30º.
cos 60° = sin 30º.
(For more on trigonometry, see the Algebra section)