In mathematics, trigonometric identities are an important class of identities that have a wide range of applications in trigonometric functions. Among them, sin cos = 1 is one of the most basic and common trigonometric identities. This article will elaborate on how to prove this identity and the mathematical principles behind it through trigonometric identity and geometric methods.
Trigonometric identity method proof
First, let's start with the definition of trigonometric functions, i.e., sin = opposite hypotenuse, cos = adjacent hypotenuse. Consider a right-angled triangle with an acute angle of , corresponding to two right-angled sides of a and b lengths, and an hypotenuse of c. According to the Pythagorean theorem, we have a +b = c.
Now, let's add sin and cos, and using the definition of trigonometric functions and the Pythagorean theorem, we have:
sin²θ cos²θ
opposite / hypotenuse)² adjacent / hypotenuse)²
opposite)² / (hypotenuse)² adjacent)² / (hypotenuse)²
opposite)² adjacent)² / (hypotenuse)²
a² +b² / c²
c² / c²
Therefore, we successfully start from the definition of trigonometric functions and the Pythagorean theorem, and derive and transform to obtain sin cos = 1.
Proof of geometric methods
Another way to prove sin cos = 1 is to perform a geometric derivation using the properties of the unit circle. A unit circle is a circle of radius 1 with the center of the circle at the coordinate origin (0,0).
Suppose is an angle on a unit circle, represented in a polar coordinate system. Then, the coordinates of the point corresponding to the angle are (cos, sin). Depending on the nature of the unit circle, the distance from the point to the origin is 1, i.e. (cos sin) = 1. Then, square both sides of the equation to get cos sin = 1 = 1.
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It is hoped that the elaboration of this article will help readers better understand and grasp the mathematical concepts of this foundation, and also show the beauty and rigor of mathematical proofs. Through in-depth analysis and derivation, we can discover the mysteries of the mathematical world, which in turn stimulates interest and desire to explore mathematics.