Supplementary mathematics/Trigonometric functions

In mathematics, trigonometric functions mean the six functions of sine, cosine, tangent, cotangent, secant, and secant, which represent the relationship between the corners and sides of a right-angled triangle, and for this reason, they are called trigonometric functions. The first surviving texts of trigonometric functions date back to BC in Egypt and Greece. The issue of Thales was raised by Thales in the sixth century BC in Egypt, and the Pythagorean theorem is also mentioned as the cornerstone of trigonometry. In addition to Egypt and Greece, there are other notable advances in the field of trigonometry, including China, India, Islam, and Europe, which can be mentioned by people such as Khwarazmi, Bettani, Abulofa Mohammad Bozjani, Shen Kuo, Shujing, and Reticus.

Arabic is expressed differently from trigonometric functions, the simplest of them is based on the unit circle, in this definition, a circle with a radius of 1 is drawn, and a radius with an angle relative to the horizontal is drawn on it and gives a triangle. Each of the trigonometric functions can be represented by a line in this circle. Other definitions of trigonometric functions are also based on integral, power series and differential equations, each of which has its own applications. For example, in the definition based on the power series, the McLorran series is used, and many trigonometric functions are used to calculate their approximate value.

Trigonometric functions operate on a point and return a real number, and each of them has its own properties, including being even or odd, alternating, continuous, orthogonal. The main use of these functions is to calculate the size of the sides and angles of a triangle and other related factors. This application is used in different sciences such as mapping, navigation and different fields of physics. In mapping, by measuring the angle of a point with respect to two given points, it calculates that point, which is used for three-dimensional optical measurement or in navigation, setting the line of ships and other vessels at the base of objects. It is fixed like a lantern. Marine is done using trigonometric functions. Also, due to the periodic characteristics of these functions, they are used in modeling oscillatory processes such as light and waves. For example, the law is the most basic application of trigonometric functions, which is used in the phenomenon of light refraction. Other applications of trigonometric functions can be mentioned in the electricity industry. including sinusoidal applications in currents as well as types of modulation based on this sinusoidal text.

Definition of principles and rules

The number of trigonometric functions originally has six basic trigonometric functions in trigonometry and is used in mathematics now. These functions are trigonometric ratios. The six main trigonometric functions are the sine function, the cosine function, the tangent function, the cotangent function, the secant function, and the cosecant function. Trigonometric functions and identities are the ratio of the sides of a right triangle. The sides of a right triangle are the perpendicular, hypotenuse, and base, which are used to calculate the values of sine, cosine, tangent, secant, cotangent, and secant simultaneously using trigonometric formulas.

Definition based on right triangle

You have probably seen the trigonometric ratios of cosine, sine and tangent in a right triangle and used them to calculate the sides and You have used the angles of those triangles.In this booklet, we review the definition of these trigonometric ratios and expand the concept of cosine, sine and tangent. We define cosine, sine and tangent as functions of all real numbers. They are assumed in most mathematics courses of the first year of university.

Sine

Sine is the first trigonometric function. It is obtained from the ratio of the opposite side to the chord.The sine function  is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let  be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then  is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.

The common schoolbook definition of the sine of an angle  in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,

$*y=\sin(x)*\sin(x)={\frac {opposite}{hypotenuse}}$

cosine

The cosine of the function cos(x) range is a trigonometric function that is obtained from the ratio of the adjacent side to the chord.

The cosine function  is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let x be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then  is the horizontal coordinate of the arc endpoint.

The common schoolbook definition of the cosine of an angle  in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.:

$*y=\cos(x)*\cos(x)={\frac {adjacent}{hypotenuse}}$

definition based on unit circle

The opposite figure shows a single circle on which the trigonometric functions of the angle θ are drawn. When the radius OA is placed on the circle with an angle θ to the horizontal axis, the value of the trigonometric functions can be obtained as the size of certain line segments. The values of the sine and cosine functions are drawn with dashed lines (red and blue, respectively) on the two main coordinate axes. In other words, the image of the line segment OA on the horizontal axis is equal to cosine θ and its image on the vertical axis is equal to sine θ. The length of the line segment (in pale brown) that extends tangent to the circle from point A to the horizontal axis is the tangent θ. The extension of the same segment from point A to the vertical axis (in orange color) also shows the cotangent θ. In the same way, the value of the secant and the secant of the angle θ can also be calculated.

In the unit circle, it is also possible to calculate trigonometric functions for angles greater than 90 degrees. The value of trigonometric functions for each angle is determined in the same way as above. The sign of a function is obtained based on the value of the angle in the unit circle based on the following table: |Estegan|1972|K=Guide to mathematical functions with relations, graphs and mathematical tables|p=73}}</ref>

Function	First quarter	Second quarter	third quarter	fourth quarter
The sine of the exponent	+	+	-	–
Cosine Template:Sahsant	+	-	-	+
Tangent Template:Seqcotangent	+	-	+	–

Era

Trigonometric functions for angles greater than 90 degrees can be obtained using the relationships of rotations about the center of the circle. Also, angles smaller than zero can be calculated with the rotation around the horizontal axis. The following table shows these relationships:

Rotation around the horizontal axis The period with an angle of π/2	Period with angle π	A period with an angle of 2π
$\sin(-\theta )=-\sin \theta$	$\sin(\theta +{\tfrac {\pi }{2}})=+\cos \theta$	$\sin(\theta +\pi )=-\sin \theta$	$\sin(\theta +2\pi )=+\sin \theta$
$\cos(-\theta )=+\cos \theta$	$\cos(\theta +{\tfrac {\pi }{2}})=-\sin \theta$	$\cos(\theta +\pi )=-\cos \theta$	$\cos(\theta +2\pi )=+\cos \theta$
$\tan(-\theta )=-\tan \theta$	$\tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta$	$\tan(\theta +\pi )=+\tan \theta$	$\tan(\theta +2\pi )=+\tan \theta$
$\cot(-\theta )=-\cot \theta$	$\cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta$	$\cot(\theta +\pi )=+\cot \theta$	$\cot(\theta +2\pi )=+\cot \theta$
$\sec(-\theta )=+\sec \theta$	$\sec(\theta +{\tfrac {\pi }{2}})=-\csc \theta$	$\sec(\theta +\pi )=-\sec \theta$	$\sec(\theta +2\pi )=+\sec \theta$
$\csc(-\theta )=-\csc \theta$	$\csc(\theta +{\tfrac {\pi }{2}})=+\sec \theta$	$\csc(\theta +\pi )=-\csc \theta$	$\csc(\theta +2\pi )=+\csc \theta$

Animation showing the generation of sine and cosine functions in Cartesian coordinates using the unit circle.

generalization

overall

A convenient mnemonic for remembering the definition of the sine, as well as the cosine and tangent, is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).

As a result of its definition, the sine function is periodic with period $2\pi$. By the Pythagorean theorem,  also obeys the identity

$\sin ^{2}(x)+\cos ^{2}(x)=1$

Of course, there are other relationships:

$\cos(x)=1+\sin(x)$

$\sin(x)=1-\cos(x)$

identities

Identity relations in trigonometric functions can also be generalized, and binomial expansion, square binomial Identity, common term Identity, conjugate Identity, etc. are established with these cases of sine, cosine, tangent and cotangent.Pay attention to these relationships:

Binomial square identity

$(\sin(x)+\cos(x))^{2}=\sin ^{2}(x)+2\sin(x)\cos(x)+\cos ^{2}(x)$

$(\sin(x)-\cos(x))^{2}=\sin ^{2}(x)-2\sin(x)\cos(x)+\cos ^{2}(x)$

Square of The Difference of Two Binomials

$(\sin(x)+\cos(x))(\sin(x)-\cos(x))=\sin ^{2}(x)-\cos ^{2}(x)$

Are you familiar with binomial expansion? Binomial expansion in algebraic expressions is generalized in this way.

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}$

Trigonometric expressions can also be generalized with binomial expansion. For example, sindes with cosines and tangent with cotangent are placed in binomial expansion.

Form is it:

$(\sin(x)+\cos(x))^{n}=\sum _{k=0}^{n}{n \choose k}\sin ^{n-k}\cos ^{k}$