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Chapter 4: Problem 10

Evaluate the expression without using a calculator. $$\arctan 1$$

Short Answer

Expert verified
\(\arctan 1 = \frac{\pi}{4}\) radian

Step by step solution

01

Understanding the arctan function

The arctan function is the inverse of the tangent function. Given an angle \(A\), \(\tan A\) is the ratio of the opposite side to the adjacent side of a right triangle. Consequently, \(\arctan x\) gives the angle whose tangent is \(x\).
02

Identify the angle

Recall a right triangle where the sides are of equal length. This triangle is known as an isosceles right triangle or a 45-45-90 triangle. From the property of this triangle, \(\tan 45^\circ = \frac{1}{1} = 1\). Therefore, the angle whose tangent is 1 is \(45^\circ\). However, in mathematics (unless specified otherwise), angles are usually measured in radians instead of degrees.
03

Convert to radians

To convert degrees to radians, use the ratio \(\pi \: rad = 180^\circ\). So, \(45^\circ = \frac{45}{180} \pi = \frac{1}{4} \pi\) radian.

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Most popular questions from this chapter

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow 1^{-}, \text {the value of } \arccos x \rightarrow\text { _____ } .$$

Data Analysis The table shows the average sales \(S\) (in millions of dollars) of an outerwear manufacturer for each month \(t,\) where \(t=1\) represents January. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Sales, } S & 13.46 & 11.15 & 8.00 & 4.85 & 2.54 & 1.70 \\\ \hline \end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Sales, } S & 2.54 & 4.85 & 8.00 & 11.15 & 13.46 & 14.30 \\ \hline \end{array}$$ (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model's amplitude in the context of the problem.

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