Problem 211 - Calculus Volume 1

Question

Evaluate the functions. Give the exact value. \(\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

Step-by-Step Solution

Answer

\( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \)

Step 1: Understand the Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1}(x) \), gives the angle \( \theta \) whose cosine is \( x \). The range of \( \cos^{-1}(x) \) is \([0, \pi]\), which means the resulting angle will be in the first or second quadrant.

Step 2: Identify the Known Cosine Value

We need to find an angle \( \theta \) such that \( \cos(\theta) = \frac{\sqrt{3}}{2} \). Look for standard angles on the unit circle.

Step 3: Recall Unit Circle Values

On the unit circle, \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \). The unit circle is a helpful tool for recalling these standard trigonometric values for common angles.

Step 4: Check Angle Range

Check that \( \frac{\pi}{6} \) falls within the range \([0, \pi]\). Since \( \frac{\pi}{6} \approx 0.5236 \) is definitely within this range, it is a valid answer for \( \cos^{-1}(\frac{\sqrt{3}}{2}) \).

Step 5: Confirm the Result

Having verified that no other angle in the range \([0, \pi]\) works with \( \cos(\theta) = \frac{\sqrt{3}}{2} \), we conclude that the exact value is correct.

Key Concepts

unit circle

The unit circle is an essential concept when working with trigonometric functions, especially inverse functions like the inverse cosine. It is a circle centered at the origin of a coordinate plane with a radius of one unit. This seemingly simple tool is quite powerful.
On the unit circle:

• The x-coordinate of any point represents the cosine of the angle formed by drawing a line from the origin to that point.
• The y-coordinate represents the sine of that angle.
• Angles are measured from the positive x-axis, counterclockwise for positive angles, and clockwise for negative angles.

The unit circle allows you to visualize and determine the cosine (and sine) of angles, making it easier to understand trigonometric values and their patterns.
What is remarkable about the unit circle is that for each standard angle, there are specific and commonly known values of cosine and sine, which are essential for solving problems involving inverse trigonometric functions.

cosine function

The cosine function is one of the fundamental trigonometric functions. Denoted by \( \cos \theta \), it relates the angle \( \theta \) to the adjacent side and hypotenuse of a right triangle. It represents the x-coordinate of a point on the unit circle corresponding to that angle.
Key points about the cosine function include:

• Its range is from -1 to 1, meaning any point along the x-coordinate of the unit circle lies within this interval.
• The function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
• When discussing the inverse cosine or \( \cos^{-1}(x) \), we focus on angles in the range \([0, \pi]\), where the function is non-decreasing, making it invertible.

The cosine function is crucial in determining the angles that correspond to specific cosine values, especially when interpreting inverse trigonometric functions.

standard angles

Standard angles are specific angles with well-known cosine and sine values. These angles are typically multiples of \( \pi/6 \), \( \pi/4 \), and \( \pi/3 \) radians, or equivalently 30°, 45°, and 60°.Some examples of standard angles are:

• \( \cos(0) = 1 \)
• \( \cos(\pi/6) = \frac{\sqrt{3}}{2} \)
• \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \)
• \( \cos(\pi/3) = \frac{1}{2} \)

These angles make it easier to calculate trigonometric functions without a calculator, enabling quick solutions to exercises like identifying an angle from a known cosine value. They are often used in conjunction with the unit circle to find exact trigonometric values.Knowing these standard angles allows for simple mental calculations and relationships that help in understanding related trigonometric concepts, especially when dealing with inverse trigonometric functions.