Problem 11 - Precalculus with Limits

Question

Evaluate the expression without using a calculator. \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

Step-by-Step Solution

Answer

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

Step 1 - Recall the definition of inverse trigonometric function

The inverse cosine function, \(\cos ^{-1}\), is the inverse function of the cosine function. It returns the angle whose cosine is a given number, between the interval [0, \(\pi\)].

Step 2 - Recognize the value of the function input

-\(\sqrt{3}/2\) is a known value of the cosine function, and it corresponds to angles of \(\pi/6\) and \(\pi/3\) in different quadrants of the unit circle.

Step 3 - Identify which angle the inverse cosine function refers to

Since the cosine of an angle is negative in the second quadrant (\(\pi/2 < x < \pi\)), the angle we are looking for corresponds to \(\pi/3\) in the second quadrant, that is \(\pi - \(\pi/3 = 2\pi/3\).

Key Concepts

Cosine Function

The cosine function is a fundamental element of trigonometry, relating the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In the context of the unit circle, which is a circle with a radius of one unit, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.

When working with the expression \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), we look for the angle belonging to \( [0, \pi] \) whose cosine gives us \( -\frac{\sqrt{3}}{2} \). This specific value corresponds to standard angles found on the unit circle, where the connection between angles and coordinates becomes a vital concept to understand.

Unit Circle

The unit circle is a powerful tool in trigonometry, charting the relationship between angles in radians and coordinates. Each point on the unit circle's circumference represents the endpoint of an angle originating at the circle's center (the origin of the coordinate system) and intersecting the circle.

In the given problem, the input \( -\sqrt{3}/2 \) is the x-coordinate on the unit circle, and it hints at an angle residing in the second quadrant, where the x-coordinate (cosine values) are negative. The unit circle helps us pinpoint this angle to \(2\pi/3\) radians, as that is where the cosine value matches the given expression.

Radians

Radians are the standard unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 equal parts, radians provide a measure that is directly related to the length of the arc of a circle. There are \(2\pi\) radians in a full circle, meaning that \(\pi\) radians encompass half the circle — the arc length equivalent to the radius multiplied by \(\pi\).

The significance of radians can be seen when evaluating the expression \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\). The solution provided is expressed in radians, \(2\pi/3\), which is a clear way to communicate the precise location of an angle on the unit circle. Understanding radians allows for a greater comprehension of the relationship between circular motion, arc length, and angles, which is essential in trigonometry and calculus.