/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Find the exact value of each exp... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 29

Find the exact value of each expression in degrees without using a calculator or table. \(\arccos (-1)\)

Short Answer

Expert verified
180镑袄迟别虫迟调掳皑

Step by step solution

01

Understand \(\text{arccos}\) function

The \(\text{arccos}\) function, also known as the inverse cosine function, gives the angle whose cosine is the given value.
02

Identify the range of \(\text{arccos}\)

The range of the \(\text{arccos}\) function is from \[0^\text{掳} \text{ to } 180镑袄迟别虫迟调掳皑\]. This means \(\text{arccos}(x)\) will always return an angle between \0^\text{掳} \text{ and } 180镑袄迟别虫迟调掳皑\.
03

Determine the cosine value for \(\text{arccos}(-1)\)

We need to find an angle \(\theta\) in degrees such that \(\text{cos}(\theta) = -1\). The cosine of \[180镑袄迟别虫迟调掳皑\] is \(-1\).
04

Conclude the value

From the previous step, since \(\text{cos}(180镑袄迟别虫迟调掳皑) = -1\), \text{arccos}(-1) = 180镑袄迟别虫迟调掳皑\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

arccos function
The **arccos function**, also known as the **inverse cosine function**, is a vital concept in trigonometry. It essentially reverses what the cosine function does. If you have a cosine value, the arccos function gives you the angle whose cosine is that value. For instance, if the cosine of an angle \(\theta\) is \(-0.5\), the arccos of \(-0.5\) will give you back \(\theta\). This is written as \(\text{arccos}(-0.5)\). Note that while the cosine function can provide multiple angles that result in the same cosine value, the arccos function provides a principal angle as its output.
  • **Important**: **arccos(x)** means the angle whose cosine is \x\.
  • **Notation**: It's written as \(\text{arccos}(x)\) or sometimes as \(\text{cos}^{-1}(x)\).
cosine value
A key to understanding **inverse trigonometric functions** is knowing how **cosine values** work. The cosine function relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. For a given angle \(\theta\), \(\text{cos}(\theta) = \frac{ \text{adjacent side} }{ \text{hypotenuse} } \). This relationship helps when working out the **arccos**. In the given problem, \(\text{arccos}(-1)\), we are looking for an angle whose cosine is \(-1\).

  • **Cosine** of \(90^\text{掳}\) is 0.
  • **Cosine** of \(0^\text{掳}\) is 1.
  • **Cosine** of \(180镑袄迟别虫迟调掳皑\) is -1.
Hence, for **arccos** of \(-1)\), the corresponding angle in this range whose cosine value matches \(-1\) is \(180镑袄迟别虫迟调掳皑\).
range of arccos function
The **range of the arccos function** is quite specific. In trigonometry, the output of the arccos function falls between \(0^\text{掳}\) to \(180镑袄迟别虫迟调掳皑\). This means that for any input within its domain (which is from -1 to 1), *arccos(x)* will always result in an angle in this interval.
  • **For \( \text{arccos}(x)\) to be defined**, \( -1 \leq \ x \leq 1. \)
  • **Resulting angle**: The angle is between **0 and 180 degrees**.
So, if you ever encounter \( \text{arccos}(x) \), remember that the result will always be a value from \(0^\text{掳}\) to \(180镑袄迟别虫迟调掳皑\).

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