Problem 28 - Algebra 2 and Trigonometry

Question

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\cos (\arcsin 1)\)

Step-by-Step Solution

Answer

The exact value is 0.

Step 1: Understand the Problem

We need to find the exact value of \( \cos(\arcsin 1) \). This involves a composition of trigonometric and inverse trigonometric functions. \( \arcsin 1 \) gives us an angle whose sine is 1.

Step 2: Determine Arcsine Value

Recall that \( \arcsin \theta \) gives an angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). The angle whose sine is 1 is \( \frac{\pi}{2} \). So \( \arcsin 1 = \frac{\pi}{2} \).

Step 3: Find the Cosine of the Angle

Now that we have found that \( \arcsin 1 = \frac{\pi}{2} \), we evaluate \( \cos(\frac{\pi}{2}) \). The cosine of \( \frac{\pi}{2} \) is 0.

Step 4: State the Exact Function Value

Thus, the exact value of \( \cos(\arcsin 1) \) is 0. This is because the trigonometric identity states that \( \cos(\frac{\pi}{2}) = 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical expressions that relate the angles and sides of right triangles. They are pivotal in simplifying and solving trigonometric expressions. These identities include well-known formulas like the Pythagorean identity:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal identities, such as \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Angle addition and subtraction formulas, among others.

These identities help to express one trigonometric function in terms of another. They are especially useful when evaluating compositions of functions, such as in the exercise involving \( \cos(\arcsin x) \). In this case, we rely on the specific identity that \( \cos\left(\frac{\pi}{2}\right) = 0 \) to find the exact value.

Arcsine Function

The arcsine function, denoted as \( \arcsin(x) \) or \( \sin^{-1}(x) \), is the inverse of the sine function. It returns the angle whose sine is the given value. The domain of \( \arcsin(x) \) is limited to \([-1, 1]\), which corresponds to the range of the usual sine function. When you input a value into the arcsine function, the output is an angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
For example, the angle whose sine is 1 is \( \frac{\pi}{2} \), because the sine function achieves its maximum value, 1, at this angle. So, \( \arcsin(1) = \frac{\pi}{2} \). Knowing this can help us solve trigonometric problems, like finding \( \cos(\arcsin 1) \), by allowing us to substitute this angle into another function.

Cosine Function

The cosine function, \( \cos(\theta) \), calculates the adjacent side over the hypotenuse in a right triangle for an angle \( \theta \). Its output is continuous and periodic, with a range of \([-1, 1]\) and a period of \(2\pi\).
Importantly, specific values at key angles often appear in exercises, such as:

• \( \cos(0) = 1 \)
• \( \cos\left(\frac{\pi}{2}\right) = 0 \)
• \( \cos(\pi) = -1 \)

In the problem \( \cos(\arcsin 1) \), we determined \( \arcsin 1 = \frac{\pi}{2} \), leading us to evaluate \( \cos\left(\frac{\pi}{2}\right) = 0 \). This showcases the cosine function's role in composite trigonometric operations, highlighting how interpreting invariant values like \( \cos\left(\frac{\pi}{2}\right) \) can simplify complex calculations.