Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arccos 0$$

Step-by-Step Solution

Answer

The exact value of \( y \) is \( \frac{\pi}{2} \).

Step 1: Understanding the Problem

We need to find the real number \( y \) such that \( y = \arccos(0) \). This means we are looking for the angle whose cosine value is 0.

Step 2: Recall the Definition of Arccosine

Recall that \( \arccos(x) \) is the angle \( \theta \) such that \( \cos(\theta) = x \), and \( \theta \) lies within the range \([0, \pi] \).

Step 3: Solving for the Angle

We need \( \theta \) such that \( \cos(\theta) = 0 \). From trigonometry, we know that the cosine of \( \pi/2 \) is 0.

Step 4: Verify the Angle is within the Range for Arccosine

Since \( \pi/2 \) falls within the range \([0, \pi] \), it is a valid solution for \( \arccos(0) \).

Key Concepts

Understanding the Concept of Arccos

The function known as the arccosine, often notated as \( \arccos \), is a fundamental concept in the world of inverse trigonometric functions. It helps us find an angle when we know the cosine of that angle. Here is how it works:

• The symbol \( \arccos(x) \) represents the angle \( \theta \) for which the cosine value is \( x \).
• This angle \( \theta \) will always be found within the range of 0 to \( \pi \) radians, corresponding to 0 to 180 degrees.

Understanding the constraints is essential because they ensure that each cosine value within the range will correspond to exactly one angle in the arccos function.
The arccosine plays a crucial role not only in mathematical calculations but also in real-world applications whenever angles need to be determined from cosine values.

Exploring the Cosine Function

The cosine function is one of the primary trigonometric functions alongside sine and tangent, and it's crucial in understanding the behavior of right-angled triangles and circles.

• The cosine of an angle \( \theta \) gives us the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
• It is also interpreted in the unit circle as the \( x \)-coordinate of a point formed by the angle \( \theta \) from the positive \( x \)-axis.
• Cosine values range from -1 to 1, repeating in a periodic pattern through complete cycles of \( 2\pi \).

When the cosine of an angle is 0, this corresponds to angles where the point lies right on the \( y \)-axis on the unit circle. For instance, \( \cos(\frac{\pi}{2}) = 0 \) is concerned with these angles.

Understanding Angle Measurement

Angles can be measured in different units, most commonly in degrees and radians. Each measurement has specific use cases, and it is important to understand how they relate.

• Degrees are often used in practical applications, and a full circle is 360 degrees. This small unit can sometimes make calculations difficult.
• Radians are more common in advanced mathematics because they simplify many formulas, particularly those related to arc length and sector area in circles.
• One full circle is \( 2\pi \) radians, thus \( \frac{\pi}{2} \) radians equate to 90 degrees.

Thus, when finding \( \arccos(0) = \frac{\pi}{2} \), we use radians for precise mathematical logic, which aligns with how angles are usually factored into trigonometric functions.