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Proving the Trigonometric Identity: sin2A cos2A 1
Proving the Trigonometric Identity: sin2A cos2A 1
In this article, we will explore a fundamental trigonometric identity: sin2A cos2A 1. We will provide a detailed proof using the definitions of sine and cosine in a right-angled triangle and the Pythagorean Theorem. This identity is crucial in trigonometry, as it helps in solving various mathematical problems and has numerous applications in fields such as physics, engineering, and more.
Introduction to Trigonometric Ratios
In a right-angled triangle ABC, where B 90°, let us call the angle A BC as A. Here, AC is the hypotenuse, the opposite side is BC, and the adjacent side is AB. The sine and cosine of angle A are defined as the ratios of the sides: sin A Opposite side / Hypotenuse and cos A Adjacent side / Hypotenuse. These definitions can be written as:
Sin A BC / AC
Cos A AB / AC
Using the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry and states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:
AB2 BC2 AC2
Dividing both sides of this equation by (AC)2, we get:
AB2 / AC2 BC2 / AC2 1
Substituting the definitions of sine and cosine, this equation can be rewritten as:
cos2 A sin2 A 1
Generalizing to Any Angle
Although we have proven the identity for angle A in a right-angled triangle, this identity generally holds for any angle. Let's consider the unit circle definition to see this more clearly. In a unit circle, where the radius is 1, the sine and cosine of an angle x are defined as the y-coordinate and x-coordinate of the point on the circle, respectively. This means:
Sin x y
Cos x x
The Pythagorean Theorem for the unit circle is similar, since the radius is 1:
x2 y2 12
Substituting the definitions of sine and cosine, we get:
cos2 x sin2 x 1
From a Right-Angled Triangle to the Unit Circle
Moving from the right-angled triangle to the unit circle, we can see that the same identity holds. If we consider a right-angled triangle ABC with B 90°, sin x a / c, and cos x b / c, where c is the hypotenuse, then:
sin2 x cos2 x (a / c)2 (b / c)2 (a2 b2) / c2
By the Pythagorean Theorem, we know that a2 b2 c2, so:
(a2 b2) / c2 c2 / c2 1
Therefore, sin2 x cos2 x 1 holds for any angle x.
Conclusion
The identity sin2A cos2A 1 is a cornerstone of trigonometry, and understanding its derivation is essential for any student of mathematics. Whether proven using a right-angled triangle or the unit circle, this identity remains a powerful tool in solving trigonometric problems and exploring deeper mathematical concepts.