Problem 29 - Algebra 2 and Trigonometry

Question

In \(24-32,\) find the exact value of each expression. $$ \cos \left(\arcsin \left(-\frac{\sqrt{3}}{2}\right)\right) $$

Step-by-Step Solution

Answer

The exact value of \( \cos \left( \arcsin \left(-\frac{\sqrt{3}}{2}\right) \right) \) is \( \frac{1}{2} \).

Step 1: Understand the Given Expression

We are asked to find \( \cos \left( \arcsin \left( -\frac{\sqrt{3}}{2} \right) \right) \). This means we first need to find an angle \( \theta \) such that \( \arcsin(\theta) = -\frac{\sqrt{3}}{2} \).

Step 2: Determine the Angle in the Correct Quadrant

The result of \( \arcsin(x) \) is always in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) (Quadrants I and IV). We need to find an angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) in the given range. This angle is \( \theta = -\frac{\pi}{3} \) because in Quadrant IV, sine values are negative, and \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \).

Step 3: Compute the Cosine of the Angle

Now that we have \( \theta = -\frac{\pi}{3} \), we need to compute \( \cos(\theta) \). Using the identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \): \[ \cos\left(-\frac{\pi}{3}\right) = \sqrt{1 - \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2}. \]

Step 4: Include the Sign of the Cosine

Finally, consider the quadrant where \( \theta = -\frac{\pi}{3} \) is located. In Quadrant IV, cosine is positive. Hence, \( \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \).

Key Concepts

Arcsin Function

The arcsin function, also known as the inverse sine function, is crucial when dealing with angles and trigonometric identities. The function, denoted as \( \arcsin(x) \), is used to find an angle whose sine value is \( x \). The range of the arcsin function is limited to \([-\frac{\pi}{2}, \frac{\pi}{2}]\), which corresponds to angles in Quadrants I and IV of the unit circle.
If given a value like \(-\frac{\sqrt{3}}{2}\), the arcsin function will provide you with an angle \( \theta \) in these quadrants where the sine of \( \theta \) equals \(-\frac{\sqrt{3}}{2}\).
This specific value, \( -\frac{\sqrt{3}}{2} \), leads us to the angle \( \theta = -\frac{\pi}{3} \), because in the fourth quadrant, the sine value is negative. Therefore, the arcsin function helps us in deducing the exact angle needed for further computations.

Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometric problems, allowing us to relate different trigonometric functions. One of the most commonly used identities is the Pythagorean identity:

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

In practical applications, once we have the sine of an angle, we can use this identity to find the cosine of the same angle, or vice versa. For example, knowing \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) from the previous section, you can find \( \cos(\theta) \) by rearranging the identity:

• \( \cos^2(\theta) = 1 - \sin^2(\theta) \)
• \( \cos(\theta) = \sqrt{1 - (-\frac{\sqrt{3}}{2})^2} \)

This results in \( \cos(\theta) = \frac{1}{2} \).
The trigonometric identities thus simplify the process of switching between different trigonometric functions, supporting a smooth problem-solving process.

Quadrant Analysis

Analyzing quadrants on the unit circle is vital to determine the signs of trigonometric functions at given angles. The unit circle is divided into four quadrants:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

For the expression \( \cos(\arcsin(-\frac{\sqrt{3}}{2})) \), once we identify the angle \( \theta = -\frac{\pi}{3} \), it's critical to know that it falls in Quadrant IV. In this quadrant, sine is negative, which was already given, and cosine is positive, leading to \( \cos(-\frac{\pi}{3}) = \frac{1}{2} \).
Understanding quadrant locations and their characteristics assist us in determining the correct sign for the function values, ensuring accuracy in trigonometric solutions.