Problem 23 - Contemporary Precalculus

Question

Use a calculator in radian mode to approximate the functional value. $$\cos ^{-1}[\cos (-8.5)]$$

Step-by-Step Solution

Answer

Answer: To find the value of the inverse cosine (arccos) of the cosine of -8.5 radians, follow these steps: 1. Use a calculator to find the cosine of -8.5 radians: $$\cos(-8.5)$$. 2. Next, use a calculator to find the arccos (or cosine inverse) of the result from Step 1: $$\cos^{-1}[\cos(-8.5)]$$. 3. The resulting value is the same angle as -8.5 radians, but it is in the range of the arccos function, which is [0, π]. This is the final answer.

Step 1: Find the cosine of -8.5 radians

Use a calculator to find the cosine of -8.5 radians: $$\cos(-8.5)$$.

Step 2: Find the arccos of the result from Step 1

Now, use a calculator to find the arccos (or cosine inverse) of the result from Step 1: $$\cos^{-1}[\cos(-8.5)]$$.

Step 3: Interpret the result

The resulting value is the same angle as -8.5 radians, but it is in the range of the arccos function, which is [0, π]. This is the final answer.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are a crucial part of understanding the relationships within trigonometry. They essentially reverse the roles of the input and output of the regular trigonometric functions. For example, while the cosine function calculates the ratio of the adjacent side to the hypotenuse for a given angle, the arc-cosine (or inverse cosine), denoted as \( \cos^{-1} \), determines an angle from that ratio.

It's important to note the range of the inverse cosine function. The result of \( \cos^{-1} \) will always be an angle in the range from 0 to \( \pi \) radians. This is because the cosine function outputs values between -1 and 1, and the inverse function must map these values back to angles within this specific interval.

Inverse trigonometric functions are particularly useful in scenarios where you need to find angles rather than sides or lengths, making them valuable tools in many fields such as physics, engineering, and computer graphics.

Radians and Degrees

Radians and degrees are two units for measuring angles, and both are commonly used. However, they represent angles in different ways. One full revolution around a circle equals 360 degrees, or \( 2\pi \) radians. Therefore, the conversion between radians and degrees is fundamental in trigonometry, allowing you to switch from one unit to the other efficiently.

To convert from degrees to radians, multiply by \( \frac{\pi}{180} \). Conversely, to convert from radians to degrees, multiply by \( \frac{180}{\pi} \). For example, 180 degrees is equivalent to \( \pi \) radians, which is a crucial benchmark for comparison.

When using trigonometric functions, be mindful of the unit setting on your calculator, especially if you're switching between radians and degrees. In this exercise, all calculations are done in radians, which is a standard practice for calculus and higher-level mathematics.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is often denoted as \( \cos \) and is used to calculate the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle for a given angle. The function itself is periodic, with a period of \( 2\pi \) radians (or 360 degrees), meaning it repeats its values in regular intervals.

Calculating \( \cos(-x) \) for any angle \( x \) involves understanding the even nature of the function. Since cosine is an even function, \( \cos(-x) = \cos(x) \). This property simplifies calculations involving negative angles, as seen in this exercise, where \( \cos(-8.5) \) simplifies the computation.

Cosine values range from -1 to 1, appearing as a smooth curve known as the cosine wave when graphed. This periodic and symmetric nature makes the cosine function very predictable and essential in various applications like signal processing and wave mechanics.