Problem 4 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \arccos (-1) $$

Step-by-Step Solution

Answer

\( ext{arccos}(-1) = \pi\).

Step 1: Understanding the Problem

We need to find the value of the inverse cosine function, also known as arc cosine, for the input (-1). The function \( ext{arccos}(x)\) returns the angle whose cosine is \(x\).

Step 2: Recognizing Domain and Range of Arccos

The function \( ext{arccos}(x)\) is defined for \(-1 \leq x \leq 1\), and it outputs values in the range \([0, \pi]\) (in radians). Check that (-1) is within the domain.

Step 3: Finding the Cosine of the Angle

To find \( ext{arccos}(-1)\), determine which angle between \(0\) and \(\pi\) has a cosine of (-1). The cosine of \(\pi\) (or 180° in degrees) is (-1), so \( ext{arccos}(-1) = \pi\).

Step 4: Verifying the Solution

Verify that \( ext{cos}(\pi) = -1\) holds true, confirming that the angle \(\pi\) is indeed the correct value for \( ext{arccos}(-1)\). This rests on the cosine property where \(\cos(\pi) = -1\).

Key Concepts

Arccosine

The term *arccosine* refers to the inverse function of cosine, frequently denoted as \(\text{arccos}(x)\). This function essentially helps us discover an angle when its cosine value is known. Unlike the regular cosine function, which takes an angle and gives us a ratio, arccosine takes the ratio and returns the corresponding angle. Understanding this distinction is crucial for mastering the concept.

• The function \(\text{arccos}(x)\) is defined specifically for inputs \(-1 \leq x \leq 1\).
• It provides an angle within the range \([0, \pi]\) when measured in radians.

In essence, if you know the cosine of an angle and seek to find the angle itself, arccosine is your go-to function. For instance, finding \(\text{arccos}(-1)\) means identifying which angle's cosine value equals \(-1\), resulting in an angle of \(\pi\), or \(180^\circ\). This clear understanding helps in solving problems involving inverse trigonometric functions.

Trigonometric Functions

Trigonometric functions are vital in mathematics. They relate angles of triangles to the lengths of their sides and are categorized primarily into sine, cosine, and tangent, along with their respective inverses: arcsine, arccosine, and arctangent. Each of these functions offers a unique perspective on circle and triangle measurements.

• Cosine, often denoted as \(\cos(\theta)\), is a trigonometric function that provides the ratio of the adjacent side to the hypotenuse in a right triangle.
• The inverse of this function, arccosine, helps determine what angle, \(\theta\), produces a known cosine value.

Inverse trigonometric functions reverse this process. While trigonometric functions take an angle and return a ratio, their inverses take a ratio and return the angle. For example, if you're given the ratio of the sides of a right triangle, these functions help you find the triangle's angle measurements. This property is central to numerous mathematical applications and calculations.

Angle Measurement

Angle measurement is pivotal in trigonometry, providing perspectives in both degrees and radians. Understanding this concept is crucial for working with trigonometric functions.
Degrees and radians are two units for measuring angles. While degrees are more common in everyday situations, radians are often preferred in mathematical contexts due to their natural relation to circles.

• A complete circle comprises \(360^\circ\) or \(2\pi\) radians.
• When dealing with functions like arccosine, the output is usually in radians. For instance, \(\text{arccos}(-1)\) is given as \(\pi\), representing half a circle or \(180^\circ\).
• Understanding these conversions helps bridge the gap between theoretical and practical applications.

Effectively interpreting these measurements ensures that you can navigate between different forms of angle expressions smoothly, enhancing your problem-solving skills in trigonometry and beyond.