Problem 40 - Algebra and Trigonometry

Question

\(23-44=\) 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 Inner Function

The expression inside the cosine function is \( \sin^{-1}(0) \). The inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), gives us the angle \( \theta \) whose sine value is \( x \).

Step 2: Solve the Inner Function

Find the angle \( \theta \) such that \( \sin(\theta) = 0 \). The basic angle that satisfies this equation is \( \theta = 0 \) because \( \sin(0) = 0 \). Thus, \( \sin^{-1}(0) = 0 \).

Step 3: Compute the Outer Function

Now, substitute \( \theta = 0 \) into the cosine function, which gives us \( \cos(0) \).

Step 4: Evaluate the Cosine Function

Use the known value \( \cos(0) = 1 \). Hence, \( \cos(\sin^{-1}(0)) = 1 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are a way to find the angle when you know the value of a trigonometric function. They are the 'reverse' functions of the standard trigonometric functions like sine, cosine, and tangent.
When we use the inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), we look for an angle \( \theta \) such that \( \sin(\theta) = x \). The range of \( \sin^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), meaning it only gives angles within this interval.

• For example: \( \sin^{-1}(0) = 0 \) because \( \sin(0) = 0 \).
• In this problem, solving the inner expression \( \sin^{-1}(0) \) gives \( \theta = 0 \).

Inverse trigonometric functions are super useful in solving trigonometric equations, especially when you need to find specific angles for given trigonometric values.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often used in mathematics to measure angles and lengths in right triangles and unit circles. It is commonly denoted by \( \cos(x) \).
The cosine of an angle is the x-coordinate of a point on the unit circle. For an angle \( \theta \), the cosine value represents the horizontal distance from the origin to the circle's perimeter.

• Especially in this exercise, after determining \( \theta = 0 \) from \( \sin^{-1}(0) \), we use \( \cos(0) \) as the final step.
• We know \( \cos(0) = 1 \).

Thus, the cosine function helps us relate an angle to a ratio of sides in right triangles or to its position on the unit circle.

Sine Function

The sine function is another essential trigonometric function, represented as \( \sin(x) \). Just like cosine, it plays a crucial role in the study of triangles and circles.
For a given angle \( \theta \), the sine function equals the y-coordinate on the unit circle. It gives the vertical distance from the origin to the point on the circle.

• In this problem, \( \sin(\theta) \) was set to 0, meaning we found the angle for \( \sin^{-1}(0) \).
• The basic angle where \( \sin(\theta) = 0 \) is \( \theta = 0 \), confirming that \( \sin^{-1}(0) = 0 \).

Understanding the sine function's properties is vital in contextualizing how inverse functions work to find angles based on specific sine values.