Problem 40 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\cos \left(\sin ^{-1} 0\right)$$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Understand the inverse sine function

The expression involves the inverse sine function, which is represented by \( \sin^{-1}(x) \). This function returns an angle whose sine value is \( x \). In this problem, you are given \( \sin^{-1}(0) \), meaning the angle whose sine is 0.

Step 2: Identify the angle whose sine is 0

The angle within the principal range \([\frac{-\pi}{2}, \frac{\pi}{2}]\) that has a sine of 0 is \( 0 \). Therefore, \( \sin^{-1}(0) = 0 \) radians.

Step 3: Find the cosine of the angle

Now that we know \( \sin^{-1}(0) = 0 \), substitute 0 for the angle in the cosine function. Thus, the expression becomes \( \cos(0) \).

Step 4: Solve for the cosine of 0

The cosine of 0 radians is 1, as \( \cos(0) = 1 \).

Step 5: Conclusion

Therefore, the exact value of the expression \( \cos(\sin^{-1}(0)) \) is 1.

Key Concepts

Inverse Sine

The inverse sine function, often denoted as \( \sin^{-1}(x) \), is a way to find the angle whose sine value equals \( x \). It's important to understand that while the sine function takes an angle and gives you a ratio, the inverse function does the opposite. It takes a ratio and gives you the angle. This is especially useful when dealing with angles within the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
In the context of the exercise, \( \sin^{-1}(0) \) means finding the angle whose sine is 0. Within the specified range, this angle is \( 0 \). The inverse sine essentially "undoes" what the sine function does, allowing you to retrieve the initial angle given the sine value.
This concept is crucial when solving problems involving angles and their respective trigonometric values.

• Inverse sine \( \sin^{-1}(x) \) finds the angle for a given sine value.
• The range for \( \sin^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Cosine Function

The cosine function is another fundamental trigonometric function closely related to sine. It is denoted by \( \cos(x) \) and provides the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function is periodic and and has a cycle every \( 2\pi \).
One of the important values to know is \( \cos(0) \), which equals 1. This reflects the fact that at \( 0 \) radians, the adjacent side length of the triangle is equal to the hypotenuse, making the ratio 1.
In our exercise, once we deduced that \( \sin^{-1}(0) = 0 \), we then find \( \cos(0) \). Remember that the cosine function at the 0 angle results in 1, which helps solve the given problem accurately.

• Cosine gives the ratio of the adjacent side to the hypotenuse.
• \( \cos(0) = 1 \), a key identity to remember.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. They are incredibly useful tools in simplifying expressions and solving trigonometric equations. Some of the fundamental identities include the Pythagorean identities, co-function identities, and angle sum and difference identities.
A relevant identity in our exercise is the Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \). This identity shows the inherent relationship between the sine and cosine of the same angle. Using this, if one function’s value is known, the other can often be deduced.
An understanding of these identities is crucial when dealing with inverse trigonometric functions and simplifying expressions, just like in the example where knowing \( \cos(0) = 1 \) directly helps in evaluating the expression.

• Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Trigonometric identities simplify expressions and solve equations.