Problem 59 - Calculus Single Variable

Question

Show that \(\arcsin (x)=\pi / 2-\arccos (x)\) for all \(x\) in [-1,1].

Step-by-Step Solution

Answer

The identity \(\arcsin(x) = \pi/2 - \arccos(x)\) is verified using complementary angles.

Step 1: Recall the Definitions of Arcsin and Arccos

The function \(\arcsin(x)\) is the inverse of the sine function on the interval \([-\pi/2, \pi/2]\), while \(\arccos(x)\) is the inverse of the cosine function on the interval \([0, \pi]\). These definitions tell us the range and domain for the respective functions.

Step 2: Use the Identity for Sin and Cos in terms of Right Triangle

For any angle \(\theta\) in a right triangle, \(\sin(\theta)\) and \(\cos(\theta)\) satisfy the identity \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\). Therefore, this can be modified to reflect the inverse functions: \(\arcsin(x) = \arccos(\frac{\pi}{2} - x)\).

Step 3: Modify the Expression using Complementary Angle Identity

From the identity \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\), by applying inverse functions, we obtain \(\arcsin(x) = \frac{\pi}{2} - \arccos(x)\) by evaluating the arcsin and arccos expressions as complementary angles.

Step 4: Confirm in the Allowed Interval

Since \(x\) is defined in \([-1, 1]\), both functions \(\arcsin(x)\) and \(\arccos(x)\) are valid across this interval due to their domain restrictions, and the identity holds true.

Key Concepts

Arcsin Function

The arcsin function, also known as the inverse sine function, is denoted as \(\arcsin(x)\). It serves as the inverse of the sine function but with a specified domain. The domain is limited to \([\-1, 1]\) to ensure the function is bijective, or one-to-one. In simpler terms, for any input \(x\) from -1 to 1, \(\arcsin(x)\) will return an angle \(\theta\) such that \(\sin(\theta) = x\).
\(\arcsin(x)\) produces an angle in the range of \([-\pi/2, \pi/2]\), which corresponds to the first and fourth quadrants of the unit circle. In these quadrants, sine values cover the full range from \(-1\) to \(1\). This range selection allows \(\arcsin\) to output angles which are continuous and strictly increasing.

Arccos Function

The arccos function, indicated by \(\arccos(x)\), works as the inverse of the cosine function. It is crucial to keep the function one-to-one, hence it is defined with a specific domain of \([\-1, 1]\). For any given \(x\) within this domain, \(\arccos(x)\) provides an angle \(\theta\) such that \(\cos(\theta) = x\).
In contrast to the arcsin function, \(\arccos(x)\) returns an angle within the range \([0, \pi]\). This interval corresponds to the first and second quadrants of the unit circle. In these quadrants, the cosine values range from \(1\) down to \(-1\). This selected range ensures that \(\arccos\) function is both continuous and strictly decreasing.

Complementary Angles

Complementary angles are two angles whose sum is \(\pi/2\) or \(90^\circ\). This concept is particularly useful in trigonometry because of the relationship it creates between sine and cosine functions.
For example, the sine of an angle is equal to the cosine of its complementary angle: \(\sin(\theta) = \cos(\pi/2 - \theta)\). This relationship directly influences the identity that \(\arcsin(x) = \pi / 2 - \arccos(x)\).

• This identity shows that for any angle \(\theta\), the sine of \(\theta\) can be described as the cosine of \(\pi/2\) minus \(\theta\), demonstrating the complementary nature of these trigonometric functions.

Understanding complementary angles helps simplify calculations in trigonometry, and it forms the basis for many trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations that hold true for any value of the variable, often seen as foundational tools in solving trigonometric problems. These identities help us understand and transform trigonometric expressions to simplify solutions.
One such identity is \(\sin(\theta) = \cos(\pi/2 - \theta)\). By applying inverse functions, we derive identities such as \(\arcsin(x) = \pi / 2 - \arccos(x)\). These relationships lead to simplified problem-solving steps and allow us to switch between different trigonometric functions with ease.

• These identities show equivalences between expressions involving different trigonometric functions.
• The use of identities simplifies complex calculations, reducing them to simpler terms.

Understanding these identities not only aids in computation but also provides deeper insights into the geometry of triangles and the unit circle.