Problem 10 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

\(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = 150^{\circ} \text{ or } 5\pi/6 \text{ rad}\)

Step 1: Understand the inverse cosine function

The inverse cosine function, or \(\cos^{-1}(x)\), will give you the angle whose cosine yields \(x\). For instance, if \(\cos(\Theta) = x\), then \(\Theta = \cos^{-1}(x)\). Its function is to return the angle between 0 and \(\pi\) (or between 0 and 180 degrees) whose cosine is \(x\).

Step 2: Associate the fraction to trigonometric value

Beginning with the specific fraction of \(-\frac{\sqrt{3}}{2}\), it is recognized as one of the well-known trigonometric ratios. However, it's negative. The cosine of 30 degrees or \(\pi/6\) rad (in radians) is \(\frac{\sqrt{3}}{2}\), but we're looking for \(-\frac{\sqrt{3}}{2}\). The cosine function is negative in the second and third quadrants.

Step 3: Identify the precise value

Because the inverse cosine function only gives values in quadrants I and II, we're looking for an angle in the second quadrant. The mirror angle of 30 degrees (\(\pi/6\) rad) in the second quadrant yields 150 degrees or \(\pi - pi/6 = 5\pi/6\) rad. That's our result.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are fundamental to understanding relationships in right-angled triangles. They are the sine, cosine, and tangent functions, which correspond to the ratios of various sides of a triangle. Each function has an inverse, allowing us to determine the angle given a ratio. Understanding these becomes crucial when solving geometry and trigonometry problems.

For example, the cosine ratio is defined as the adjacent side over the hypotenuse in a right-angled triangle. But when the cosine value is negative, as in the exercise, it indicates that we are dealing with an angle in either the second or third quadrant, where the cosine values are negative. Recognizing these patterns can help students quickly locate which quadrant they should consider for their answers and avoid common mistakes.

Trigonometry Step by Step Solution

Tackling trigonometry problems can seem overwhelming at first, but breaking down the process into steps can simplify the task. A step by step solution involves understanding the problem, identifying known values, and applying trigonometric identities and relationships. Let's use our problem as an example.

In the exercise given, the step by step solution involves first understanding the inverse cosine function—this tells us that we're looking for an angle. Next, we associate our given fraction with a known cosine value and remember that our result must be between 0 and \(\pi\) since we're working with the inverse cosine function. Lastly, identify the accurate quadrant for the angle using the sign of the ratio, allowing us to find the precise value. These structured steps can turn a complicated process into a more manageable one.

Cosine of an Angle

The cosine of an angle is a trigonometric function that compares two sides of a right-angled triangle. Specifically, it is the length of the adjacent side divided by the length of the hypotenuse. When we're given the cosine of an angle, we're often asked to work with the inverse of this function, denoted by \(\cos^{-1}\).

This inverse function helps us find the angle when we have the side lengths. However, as angles can be in any of the four quadrants, it's essential to remember that the cosine function can be positive or negative. In the context of our exercise, we used the negative value to determine that our angle must be in the second quadrant and, therefore, deduced the correct angle as 150 degrees or \(5\pi/6\) radians. Understanding this is critical in solving trigonometry problems accurately.