Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\arcsin 0\) (b) \(\arccos (-1)\) (c) \(\arctan 0\)

Step-by-Step Solution

Answer

(a) 0, (b) \(\pi\), (c) 0.

Step 1: Understanding Arcsin

The arcsin function or inverse sine function \(\arcsin x\) returns the angle whose sine is \(x\). For \(\arcsin 0\), we need an angle where the sine value is zero. The angle that satisfies \(\sin(\theta) = 0\) within the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) is \(\theta = 0\).

Step 2: Finding Value for Arccos

The arccos function or inverse cosine function \(\arccos x\) returns the angle whose cosine is \(x\). For \(\arccos(-1)\), we need an angle where the cosine value is -1. The only angle that satisfies \(\cos(\theta) = -1\) within the range \(0 \leq \theta \leq \pi\) is \(\theta = \pi\).

Step 3: Understanding Arctan

The arctan function or inverse tangent function \(\arctan x\) returns the angle whose tangent is \(x\). For \(\arctan 0\), we need an angle where the tangent value is zero. The angle that satisfies \(\tan(\theta) = 0\) within the range \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) is \(\theta = 0\).

Key Concepts

Arcsin

The arcsin function is also known as the inverse sine function. It's denoted as \( \arcsin x \). This function tells us what angle, within a specific range, has a sine value equal to \( x \). For the arcsin function, the range is set between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This is important because it helps keep the function a true inverse, matching every sine value in the range to exactly one unique angle.

When we solve \( \arcsin 0 \), we're looking for an angle \( \theta \) such that \( \sin(\theta) = 0 \). Within our range, the angle that makes the sine of zero is \( \theta = 0 \).

• This range ensures that each value of \( x \) produces exactly one angle \( \theta \).
• Remember, the sine of zero is always zero, which is why \( \arcsin 0 = 0 \).

Arccos

Arccos, or the inverse cosine function, is represented as \( \arccos x \). This function finds the angle in the range from \(0\) to \(\pi\) whose cosine is \( x \). Cosine values can range from -1 to 1, and arccos helps identify the specific angle that corresponds to a given cosine value within the primary range.

For \( \arccos(-1) \), we seek an angle \( \theta \) where \( \cos(\theta) = -1 \). The angle within our range \([0, \pi]\) that fits this requirement is \( \theta = \pi \).

• The only angle in this range where cosine equals -1 is \( \pi \).
• This means \( \arccos(-1) = \pi \), giving us a straightforward solution.

Arctan

Arctan, or the inverse tangent function, is denoted by \( \arctan x \). This function helps us find the angle whose tangent is \( x \). The range for the arctan function is a bit different—it stretches from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), but note that these bounds do not include the endpoints.

In the case of \( \arctan 0 \), we aim to identify an angle \( \theta \) such that \( \tan(\theta) = 0 \). The angle \( \theta \) where this condition holds true, within our specified range, is \( 0 \).

• The tangent of zero equals zero, making \( \arctan 0 = 0 \).
• This solution works perfectly within the specified range: \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).