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Chapter 5: Problem 5

Find the exact value of the given expression in radians. \(\cos ^{-1}(-1)\)

Short Answer

Expert verified
\(\pi\)

Step by step solution

01

Understand the Problem

We need to find the angle whose cosine is \(-1\). This angle will be expressed in radians.
02

Recall the Cosine Graph

Recall the graph or unit circle representation of the cosine function, which ranges from \[ -1 \] to \[ 1 \] and completes one cycle over \[ 0 \ to \ 2\pi \].
03

Identify the Special Angle

On the unit circle, the cosine of \(\pi\) is \(-1\).
04

Verify the Answer

Since \(\pi\) is within the range \[0, \pi\] for the inverse cosine function, the answer is validated.

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Key Concepts

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

Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a fundamental tool in trigonometry for understanding angles and their corresponding sine and cosine values.

Here鈥檚 why it鈥檚 so useful:
  • Angles are measured in radians.
  • Every point on the circle has coordinates \((\text{cos}(\theta), \text{sin}(\theta))\).
For example, at an angle of \(0\) radians (or 0 degrees), the coordinates are \((1, 0)\). When you reach \(\frac{\text{pi}}{2}\) radians (90 degrees), the coordinates are \((0, 1)\).

To find the \(\text{cos}^{-1}(-1)\), we look at when the cosine value on the unit circle is -1. This happens at \(\text{pi}\) radians, where the coordinates are \((-1, 0)\).
Cosine Graph
The graph of the cosine function is another key part of understanding trigonometry. This graph helps us easily identify the cosine values at various angles.

Here are some aspects of the cosine graph:
  • The cosine function oscillates between -1 and 1.
  • It completes a full cycle every \(2\text{pi}\) radians.
  • The graph is symmetric about the y-axis.
To find where \(\text{cos}^{-1}(-1)\) occurs, we look for where the cosine value is -1. From the cosine graph, we see that this happens at \(\text{pi}\) radians. The graph dips to -1 and confirms that \(\text{pi}\) is the correct answer.
Radians
Radians are a way of measuring angles based on the radius of a circle. This is different from degrees, which divide a circle into 360 parts. Here鈥檚 why radians are useful:
  • They help in simplifying and unifying mathematical formulas.
  • \text{pi} radians is equivalent to 180 degrees.
  • The circumference of a circle is \(2\text{pi} r\) which relates directly to \(2\text{pi} \) radians in a circle.
For the problem \(\text{cos}^{-1}(-1)\), we use radians. Since \(\text{cos}(\text{pi}) = -1\), we find that the value of \(\text{cos}^{-1}(-1)\) is \(\text{pi}\) radians. This is within the range of the inverse cosine function, which is \([0, \text{pi}]\).

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Most popular questions from this chapter

Find the period of the given function. Graph one cycle using Gnuplot. \(y=\sin \frac{x}{3}+2 \cos \frac{3 x}{4}\)

Use Gnuplot to graph the function \(y=\frac{1}{x^{2}} \sin 80 x\) from \(x=0.2\) to \(x=\pi\). What functions form its amplitude envelope? (Note: Use set samples 500 in Gnuplot.)

Find the amplitude, period, and phase shift of the given function. Then graph one cycle of the function, either by hand or by using Gnuplot (see Appendix B). \(y=-\sin (5 x+3)\)

Find the exact value of the given expression in radians. \(\tan ^{-1}\left(\tan \left(-\frac{5 \pi}{6}\right)\right)\)

Find the exact value of the given expression in radians. \(\cos \left(\sin ^{-1}\left(\frac{5}{13}\right)\right)\)
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