The expression $\arcsin(\cos{x})$ is a common problem encountered in calculus, particularly in trigonometry and function analysis. It often appears in exercises like those found in Spivak's Calculus, notably in Chapter 15, problem 18. This expression represents the angle whose sine is equal to the cosine of another angle x. The goal is to find a simpler form of this expression, which we can achieve through the use of trigonometric identities and understanding the behavior of the $\arcsin$ and $\cos$ functions.
Understanding the Problem
Let's break down what $\arcsin(\cos{x})$ means.
- arcsin: The arcsine function, denoted as $\arcsin$ or $\sin^{-1}$, is the inverse of the sine function. It takes a value between -1 and 1 (the range of the sine function) and returns the angle whose sine is that value. For example, $\arcsin(1/2) = \pi/6$ because $\sin(\pi/6) = 1/2$.
- cos: The cosine function is a trigonometric function that takes an angle and returns the ratio of the adjacent side to the hypotenuse in a right triangle.
- $\arcsin(\cos{x})$: This expression means we're taking the cosine of an angle x, and then finding the angle whose sine is equal to that cosine value.
Solving the Problem
To simplify $\arcsin(\cos{x})$, we can use a few key trigonometric identities and the properties of the arcsine function.
1. Using the Cosine Identity
We can express the cosine of an angle in terms of the sine of its complementary angle:
cos(x) = sin(π/2 - x)
2. Substituting the Identity
Now we can substitute this identity into our original expression:
arcsin(cos(x)) = arcsin(sin(π/2 - x))
3. Using the Inverse Property
The arcsine function is the inverse of the sine function. This means that:
arcsin(sin(y)) = y
Applying this to our expression:
arcsin(sin(π/2 - x)) = π/2 - x
4. Finding the Range
The arcsine function has a restricted range of -π/2 to π/2. We need to ensure our solution stays within this range.
- If x is between 0 and π/2, then π/2 - x will also be between 0 and π/2, and our solution is valid.
- If x is between π/2 and π, then π/2 - x will be between -π/2 and 0. We need to adjust our solution by adding 2π: π/2 - x + 2π.
The Solution
Therefore, the simplified form of $\arcsin(\cos{x})$ depends on the value of x:
- For 0 ≤ x ≤ π/2: arcsin(cos(x)) = π/2 - x
- For π/2 ≤ x ≤ π: arcsin(cos(x)) = 3π/2 - x
Visual Representation
The relationship between $\arcsin(\cos{x})$ and the original angle x can be visualized using the unit circle.
Figure 1: Unit Circle Representation
[Insert a diagram of the unit circle here, illustrating the relationship between x and π/2 - x, and how their cosines and sines relate.]
In the unit circle, the cosine of an angle is represented by the x-coordinate, while the sine is represented by the y-coordinate. As you can see from the diagram, the angle x and its complementary angle (π/2 - x) have the same cosine value. This directly demonstrates the relationship between $\arcsin(\cos{x})$ and the original angle x.
Importance of Range
The restriction of the range of the arcsine function is crucial in determining the correct solution. Without it, we could get multiple angles that satisfy the equation. The range ensures that we obtain a unique and specific solution for each input value of x.
Conclusion
Understanding the properties of trigonometric functions and their inverses is essential for solving expressions like $\arcsin(\cos{x})$. Through the use of identities and the range restriction, we can simplify the expression and find a solution within the defined domain. Problems like this highlight the importance of mastering trigonometric concepts and their applications in calculus. By carefully considering the range and using appropriate identities, we can confidently simplify complex trigonometric expressions and deepen our understanding of these important mathematical tools.
