Angles have more value than you can imagine. What do you not believe? Well, we encourage you to read this post where we will explain what complementary angles are – Definition, Meaning and Concept, so that you can learn a lot about the subject.
Trigonometry begins in ancient Babylon and Egypt; although later it would develop in classical Greece, Arabia and the West.
Angle is a word that has an etymological origin from the Greek and it is derived from the word “Ankulos” which means “Twisted”. This definition is very necessary to know what complementary angles are from now on.
But to deepen the understanding of trigonometry, which is the science that is responsible for the study of angles, we have developed a historical synthesis, and we will tell you below.
What are the beginnings you need to know about the development of trigonometry?
Already in Babylon more than 3000 years ago angles were used for various activities, including agriculture; over the centuries the Egyptians would use them in the construction of their wonderful pyramids.
The latter would also use the angles to calculate the position of the stars and their orbits, prepare the calendar and establish the positions of the navigation routes.
Thanks to the angles and their study, the Egyptians would discover interesting phenomena such as the calculation of time, which was essential for the organization and development of civilization. Centuries later this knowledge would be transferred to classical Greece.
Already in the second century, BC in Greece, the mathematician and astronomer Hipparco de Nicaea would be in charge of developing the science of Trigonometry, this mathematician would elaborate the table of chords to solve flat triangles. This table would be in charge of relating the measures of the angles with the measures of length, which would consist of a half circle that went from 0 to 180 degrees; this is what is now known as the sine table.
The Greeks would also adopt the sexagesimal number system from Babylon, and it is from there that 300 years later the astronomer Alejandrino Ptolemy would use the radius = 60 degrees. Ptolemy would also create the “Menelaus Theorem” which was widely used to solve spherical triangles. He would also apply his trigonometric theories to the development of astrolabes and sundials.
Already in the 8th century, it was the Arab astronomers who continued the inherited work on the studies of trigonometry that the Greeks would develop. They would begin in that century to expand the sine function; Already in the 10th century they would have further developed the trigonometric functions of cosine tangent, cotangent, secant and cosecant.
The fundamental theorems of trigonometry for plane, spherical, and polar triangles were successfully developed in this period. The Arabs also demonstrated. They also discovered the value of radius=1.
All these discoveries would be used in astronomy studies, managing to create astronomical time, with which they would find the direction of Mecca.
They also compiled the sine and tangent tables, built with intervals of 1/60 of a degree (1 minute), which had an error of less than 1 divided by 700 million.
The German mathematician and astronomer known as Regiomontano, would carry out a relevant work on trigonometric functions called “De Triangulus”. This would develop in the 12th century in the West, through translated manuscripts from Arab culture.
In the 13th century, another German astronomer known as Reticus would develop and modernize the trigonometric functions of proportions.
In the 16th century, the mathematician François Viete from France, would introduce in his work “Canon mathematics”, the notions of polar triangle and would also discover formulas for functions of multiple angles as a function of powers and as a function of simple angles.
Since then these discoveries of trigonometry as a study of circular lines and the algebra of polynomials have been used until today.
Instructions
Below we leave you everything you need to know about the characteristics, utility and way of calculating complementary angles:
- To understand what complementary angles are, you must know that the angle is one that is formed by two lines, which intersect each other in the same plane.
- The region of the plane comprised by two rays that have the same vertex, is what is known as an angle.
- From the measures of the angles that are in degrees, the classifications of the same are derived.
- Complementary angles are those angles that together form 90 degrees.
- The complementary angles added together are equal to the value of the right angle, which is 90 degrees.
- Complementary angles will always add up to 90 degrees, whether or not they are consecutive.
- Even if they are not together, if two angles add up to 90 degrees, then they are complementary angles.
- These complementary angles, as we said before, have the same measure as a right angle.
- Complementary angles consist of two sides and a vertex at the origin of each.
- Complementary angles are widely used in architecture and civil engineering.
- Complementary angles can be found in many physical phenomena.
- Two complementary angles do not need to be adjacent angles.
- Complementary angles can form various trigonometric relationships.
- One of these relationships is the one that is found in the sum of the internal angles of a right triangle that gives 180 degrees, which is made up of two complementary acute angles plus an angle of 90 degrees.
- The complementary angles are obtained in the following way: as we already know, the complementary angles are those that add up to 90 degrees or π/2 rad.
- If we have two angles, α = 40⁰ and β = 50⁰, performing the sum we have that gives 90 degrees, therefore they complement each other.
- If we want to calculate the complementary angle of a specific angle, we only have to subtract 90 degrees from the known angle. Example of this we have: 90 – 65=25 degrees, the resultant is the complementary angle.
Or what is the same, a right angle is taken as a reference and the first angle to which the complement is sought is subtracted. - Adjacent complementary angles are known as those that have the same vertex, and that added together give 90 degrees.
- The diagonal of a rectangle is made up of complementary angles (90°) with adjacent sides.
- An interesting fact is that light forms non-consecutive complementary angles through a lens.
- About the trigonometric functions of complementary angles we have that they have been established in order to expand the various functions of real and complex numbers.
- These trigonometric functions of complementary angles are essential in sciences such as mathematics, physics , astronomy, cartography, and telecommunications.
- They are defined as the quotient that exists between the sides of a right triangle and its relationship with the angles.
- It is interesting to add that supplementary angles are those angles that together add up to 180 degrees.
And that in addition a flat angle also adds, 180 degrees.
Tips
The trigonometric ratios of the complementary angles are those that can be obtained as a function of the trigonometric ratios of α.
Here we will leave these functions so that you know how to use them when making the respective calculations:
- Sine:
sin (90° – α) = cos α
Cosine:
cos (90° – α) = sin α
Tangent:
Tan (90° – α) = cot α - Cosecant:
csc (90° – α) = sec α
Secant:
sec (90° – α) = csc α - Cotangent:
cot (90° – α) = tan α