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Chapter 4: Problem 7

Evaluate the expression without using a calculator. $$\arccos \frac{1}{2}$$

Short Answer

Expert verified
\(\arccos \frac{1}{2} = \frac{\pi}{3}\) or \(60^o\)

Step by step solution

01

Understanding Arccosine Function

The arccosine function, denoted as \(\arccos(x)\) or \(cos^{-1}(x)\), is the inverse of the cosine function. It returns the angle whose cosine is \(x\). It's also important to know that the range of \(y = cos(x)\) is \([-1, 1]\) and the range of \(x = \arccos(y)\) is \([0, \pi]\) or \([0, 180^o]\) in radians or degrees respectively.
02

Finding the Angle

We're asked to evaluate \(\arccos \frac{1}{2}\). From the unit circle or standard triangles like the 30-60-90 triangle, we know that the cosine of \(\frac{\pi}{3}\) or \(60^o\) is \(\frac{1}{2}\).
03

Writing Final Answer

So, \(\arccos \frac{1}{2} = \frac{\pi}{3}\) or \(60^o\).

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Most popular questions from this chapter

Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)\) 3 inches Amplitude 3 inches Period 1.5 seconds

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=9 \cos \frac{6 \pi}{5} t$$

The circular blade on a saw rotates at 5000 revolutions per minute. (a) Find the angular speed of the blade in radians per minute. (b) The blade has a diameter of \(7 \frac{1}{4}\) inches. Find the linear speed of a blade tip.

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin (x-1)$$
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