Problem 247 - Precalculus

Question

For the following exercises, find the exact value. $$ \cos ^{-1}(\sin (\pi)) $$

Step-by-Step Solution

Answer

The exact value is \(\pi/2\).

Step 1: Evaluate the inner function

First, evaluate the expression inside the parentheses, \( \sin(\pi) \). From trigonometric identities, we know that \( \sin(\pi) = 0 \). This simplifies the expression to \( \cos^{-1}(0) \).

Step 2: Determine the angle for \( \cos^{-1}(0) \)

Next, find the angle whose cosine value is 0. The inverse cosine function \( \cos^{-1}(x) \) gives the angle in the principal range \([0, \pi]\). In this range, \( \cos(\pi/2) = 0 \). Therefore, \( \cos^{-1}(0) = \pi/2 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables, wherever the expressions are defined. They help simplify complex trigonometric expressions. One of the key identities to remember is the

• Sine of 0 degrees and 180 degrees, which are 0.
• Cosine of 90 degrees, which is also 0.

These identities are not only helpful in simplifying expressions but are crucial in solving trigonometric equations too. In our original exercise, we used the identity \(\sin(\pi) = 0\) to simplify the problem. Recognizing these trigonometric identities allows us to evaluate functions accurately and efficiently.

Inverse Cosine Function

The inverse cosine function, denoted as \(\cos^{-1}(x)\), is a function used to find angles when the cosine value is known. Inverse trigonometric functions can reverse the action of trigonometric functions, with specific ranges. For \(\cos^{-1}(x)\), this function provides an angle whose cosine is \(x\). The range of this function is particularly important because:

• It only returns angles within \([0, \pi]\).
• This range is part of ensuring that each cosine input maps to a unique angle output.

In problems like our exercise, where we start with a cosine value (like 0), \(\cos^{-1}(x)\) can directly give us the corresponding angle in this range, like \(\frac{\pi}{2}\) for \(\cos^{-1}(0)\). Understanding this function is key to solving inverse trigonometry problems effectively.

Principal Range of Cosine

The principal range of cosine is a concept that defines the range of angles that the inverse cosine function will consider. This ensures the function only provides valid outputs. For cosine, this principal range is

• \([0, \pi]\), which contains all possible angles where cosine values transition from 1 to -1.
• This makes it unique in trigonometric functions since it only includes angles from the first and second quadrants.

Knowing about the principal range is crucial because it allows us to specify the correct angle when calculating using the inverse cosine function. For instance, if \(\cos^{-1}(0)\) is calculated, this principal range tells us the answer has to be within \([0, \pi]\), leading us to \(\frac{\pi}{2}\). The principal range is part of what ensures consistent and accurate problem solving in trigonometry.