Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 71-76: Verify the identity. $$ \arccos x+\arccos \sqrt{1-x^{2}}=\frac{\pi}{2}, 0 \leq x \leq 1 $$

Step-by-Step Solution

Answer

The identity is verified; both sides equal \( \frac{\pi}{2} \).

Step 1: Understand the Problem

We need to verify the identity \( \arccos x + \arccos \sqrt{1-x^2} = \frac{\pi}{2} \) for \( 0 \leq x \leq 1 \). This involves showing that the sum of the two inverse cosine terms equals \( \frac{\pi}{2} \).

Step 2: Use Inverse Trigonometric Properties

Recall the identity for inverse trigonometric functions: \( \arccos a + \arcsin a = \frac{\pi}{2} \). Notice that \( \sqrt{1-x^2} \) can be viewed as \( \sin \theta \) when \( \cos \theta = x \). Thus, \( \arccos \sqrt{1-x^2} = \arcsin x \).

Step 3: Substitute and Simplify

Substitute \( \arcsin x \) for \( \arccos \sqrt{1-x^2} \) in the original equation:\[ \arccos x + \arcsin x = \frac{\pi}{2} \].

Step 4: Verify the Identity

We know from trigonometric identities that \( \arccos x + \arcsin x = \frac{\pi}{2} \) for \( 0 \leq x \leq 1 \). This confirms that the equation holds true for the given domain.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help in finding angles from the given trigonometric values. This is the opposite operation of the regular trigonometric functions which give us values (ratios) from angles. Some common inverse trigonometric functions include:

• Inverse Sine: \( \arcsin(x) \)
• Inverse Cosine: \( \arccos(x) \)
• Inverse Tangent: \( \arctan(x) \)

^{-1}The way inverse functions work is by answering the question of which angle results in a particular trigonometric ratio. For instance, if \(x \) represents the sine of an angle, \( \arcsin(x) \) gives back that specific angle.

Arccos Function

The \( \arccos \) function, also known as the inverse cosine function, is used to determine the angle whose cosine is a given number. It can be denoted as \( \arccos(x) \).

The range of \( \arccos(x) \) is between 0 and \( \pi \) radians (or 0 and 180 degrees). This range is chosen so that each input corresponds to only one output, making \( \arccos \) a true function. When you calculate \( \arccos(0.5) \), it gives the angle whose cosine is 0.5, and that angle is \( \frac{\pi}{3} \) radians.

• Formula: If \( \cos(\theta) = x \), then \( \theta = \arccos(x) \).
• Range: \(0 \leq \theta \leq \pi \)
• Domain: \(-1 \leq x \leq 1 \)

Arcsin Function

The \( \arcsin \) function is the inverse of the sine function. It helps us find an angle whose sine is a given value, represented as \( \arcsin(x) \).

The range for \( \arcsin(x) \) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians (or -90 to 90 degrees). This limited range ensures that each value of \(x\) produces a unique angle, thus making it a valid function. For example, \( \arcsin(0.5) \) returns \( \frac{\pi}{6} \) radians because the sine of \( \frac{\pi}{6} \) is 0.5.

• Formula: If \( \sin(\theta) = x \), then \( \theta = \arcsin(x) \).
• Range: \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \)
• Domain: \(-1 \leq x \leq 1 \)

Domain of Trigonometric Functions

The domain of trigonometric functions refers to the set of all possible values that can be input into the function to yield a valid output.### For Inverse Trigonometric Functions:

• **Inverse Sine (Arcsin)**: The domain is \( -1 \leq x \leq 1 \)
• **Inverse Cosine (Arccos)**: The domain is \( -1 \leq x \leq 1 \)
• **Inverse Tangent (Arctan)**: The domain is all real numbers

Understanding the domain is crucial because it defines which values are permissible for these functions. Typically, the domain is restricted to ensure that each angle returns a single, unique answer, maintaining the function's validity and its practical use in calculations.