Exploring the Mathematical Identity: Arcsin(θ) Arccos(θ) π/2

The identity arcsin(x) arccos(x) π/2 holds for all values of x in the interval [-1, 1], a fascinating relationship between inverse trigonometric functions. This article provides a detailed explanation of why this identity is true, and how it reflects the fundamental properties of trigonometric functions and their inverses.

Definitions and Notation

In the context of this identity, we need to define and understand the inverse sine and cosine functions, arcsin(x) and arccos(x), respectively.

Arcsin(θ)

The function arcsin(x) is the inverse sine function. It returns the angle θ such that sin(θ) x, with θ in the interval [-π/2, π/2]. This means that arcsin(x) will always return an angle within this range.

Arccos(θ)

The function arccos(x) is the inverse cosine function. It returns the angle φ such that cos(φ) x, with φ in the interval [0, π]. This implies that arccos(x) will always return an angle within this range.

Complementary Angles and the Identity

The relationship between arcsin and arccos can be understood through the concept of complementary angles. For any angle θ in the range of arcsin, the sine of that angle is x. Therefore, we can express this as:

θ  arcsin(x) amp;#8594; sin(θ)  x

Using the Pythagorean identity, we know that the cosine of the complementary angle, which is (π/2 - θ), can be expressed as:

cos(π/2 - θ)  sin(θ)  x

This relationship implies that:

π/2 - θ  arccos(x)

Since arccos(x) gives the angle whose cosine is x, we can substitute and obtain:

θ  arcsin(x) amp;#8594; π/2 - arcsin(x)  arccos(x)

Rearranging this equation, we get:

arcsin(x)   arccos(x)  π/2

Conclusion

Therefore, the identity arcsin(x) arccos(x) π/2 is true for all x in the domain of arcsin and arccos, which is [-1, 1]. This identity reflects the complementary nature of the sine and cosine functions and their inverse relationships.

To summarize, the identity arcsin(x) arccos(x) π/2 is a beautiful manifestation of trigonometric and inverse trigonometric properties.

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