Problem 41 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Understand the Problem

We need to find the exact value of \( \cos(\sin^{-1}(\frac{\sqrt{3}}{2})) \). This requires finding the value of the cosine given an inverse sine function is applied to \( \frac{\sqrt{3}}{2} \).

Step 2: Determine the Angle for Inverse Sine

The expression \( \sin^{-1}(\frac{\sqrt{3}}{2}) \) represents an angle \( \theta \) such that \( \sin(\theta) = \frac{\sqrt{3}}{2} \). The angle \( \theta \) needs to be within the range of \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). The corresponding angle is \( \frac{\pi}{3} \) radians because \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).

Step 3: Compute the Cosine of the Angle

Now that we have \( \theta = \frac{\pi}{3} \), we need to evaluate \( \cos(\theta) \). From trigonometric identities, \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).

Step 4: State the Final Answer

The value of the original expression \( \cos(\sin^{-1}(\frac{\sqrt{3}}{2})) \) is \( \frac{1}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships involving trigonometric functions that hold true for all valid angles. Knowing these identities is crucial for simplifying expressions and solving trigonometric equations. In the given problem, the ability to switch from one trigonometric function to another relies heavily on these identities. For example, the identity "Pythagorean identity" is one such key tool, which expresses the relationship between sine and cosine:

• \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]

This identity helps in expressing cosine in terms of sine, or vice versa. When given \( \sin(\theta) = \frac{\sqrt{3}}{2} \), using the above identity, we can find the cosine of \( \theta \). Knowing identities like these allows us to transition seamlessly between different trigonometric functions, which is vital in the solution of the problem where we find that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Understanding and applying these identities simplifies the process of evaluating expressions containing inverse trigonometric functions.

Right Triangle Trigonometry

Right triangle trigonometry involves understanding and using the relationships between the angles and sides of right triangles to solve problems. When dealing with inverse trigonometric functions, there is often an implicit connection to right triangle setups. This is integral in graphical interpretations and comprehending the functions.

• The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
• Similarly, the cosine of an angle is the ratio of the adjacent side to the hypotenuse:\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

In the problem, \(\theta\) was determined to be \( \frac{\pi}{3} \) because \( \sin(\theta) = \frac{\sqrt{3}}{2} \). Imagining a right triangle with these values helps reinforce why \( \cos(\frac{\pi}{3}) \) is \( \frac{1}{2} \). This visualization is not just a conceptual tool but a tangible guide for understanding relationships and evaluating trigonometric expressions.

Angle Evaluation

Angle evaluation is the process of determining the correct angles that satisfy specific trigonometric conditions. This is particularly important when working with inverse trigonometric functions, as they often represent angles whose trigonometric function corresponds to a given value. In the exercise, we focus on evaluating the result of the inverse sine function:

• Given \( \sin^{-1}(\frac{\sqrt{3}}{2}) = \theta \), the task is to find \( \theta \) in a way that meets the condition \( \sin(\theta) = \frac{\sqrt{3}}{2} \).
• The range of \( \sin^{-1} \), which is \( [ -\frac{\pi}{2}, \frac{\pi}{2} ] \), restricts \( \theta \) to areas where the sine graph is increasing.

For \( \sin(\theta) = \frac{\sqrt{3}}{2} \), the angle \( \theta = \frac{\pi}{3} \) is chosen since it falls within this interval. Through accurate angle evaluations, we correctly determine that the expression \( \cos(\sin^{-1}(\frac{\sqrt{3}}{2})) \) simplifies to \( \frac{1}{2} \), illustrating the importance of understanding both the value and orientation of angles in solving trigonometric problems.