Problem 51 - Precalculus: A Concise Course

Question

Use the properties of inverse trigonometric functions to evaluate the expression. $$ \cos [\arccos (-0.1)] $$

Step-by-Step Solution

Answer

The value of \( \cos [\arccos (-0.1)] \) is -0.1.

Step 1: Identification of the property

The formula to apply is \( \cos (\arccos x) = x\), where x is any real number.

Step 2: Apply the property

Here, we replace x with the value given in the problem. Therefore we find that \( \cos (\arccos (-0.1)) = -0.1\).

Key Concepts

Understanding Arccos

The function 'arccos' stands for the inverse of the cosine function. It's used when you want to find the angle whose cosine value is a given number. It can seem mystical at first, but once you grasp how it undoes what cosine does, it becomes quite logical.

• When you input a value into an arccos function, it tells you which angle in the interval from 0 to \(\pi\) radians corresponds to it.
• This function is essentially the reverse operation of a cosine."

Just imagine it like a backtracking tool that tracks you back from a number to its angle. That's what arccos is all about! For example, if you know that the cosine of an angle is -0.1, then \(\arccos(-0.1)\) fetches the angle whose cosine is exactly -0.1. This makes it super handy when working with circles or rotations in geometry.

Breaking Down Cosine

Cosine is one of the primary trigonometric functions, often abbreviated as 'cos'. It's all about the relationship between an angle in a right triangle and its sides. Specifically, the cosine of an angle is the ratio of the length of the adjacent side to that of the hypotenuse. In the context of the unit circle, things get even more interesting! When visualizing angles as points on a circle with radius 1 (unit circle), the cosine of an angle is the x-coordinate of that point.

• Imagine you're at the center of the circle, looking outwards along the radius – cosine is the horizontal reach in that direction.
• More practically, if you know the angle, using cosine helps you figure out this horizontal distance.

It's also important to remember that cosine values can range from -1 to 1, mapping out the entire circle in terms of horizontal width with ease. This makes it a vital part of trigonometry and essential for relating angles to their coordinates.

Trigonometric Identities Made Simple

Trigonometric identities are like the secret rules that govern the behavior of trigonometric functions. They allow us to simplify expressions, prove equations, and solve complex problems with elegance.The identity used in the original exercise is one of the inverse properties: \(\cos(\arccos(x)) = x\). This identity says that if you take the arccos of a number and then the cosine of the result, you get that same number back. Here's why:

• The arccos function gives you the angle whose cosine is your number.
• When you apply cosine to that angle, you return to your starting value.

It’s akin to retracing your steps through math! This property is part of a broader set of inverse trigonometry rules that you'll often see in problems, acting as powerful tools for simplification. Once you familiarize yourself with these identities, solving trigonometric problems becomes significantly more straightforward, keeping complexity at bay.