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Chapter 1: Problem 88

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. $$ \tan \left(-\frac{25 \pi}{7}\right) $$

Short Answer

Expert verified
To find the correct answer, use a calculator to compute the \(\tan(-\frac{3 \pi}{7})\) and round the result to 4 decimal places.

Step by step solution

01

Bringing the argument into the interval [-蟺/2, 蟺/2]

The first step is to ensure the argument to the tangent function falls within the usual interval for the definition of the trigonometric functions, \([-蟺/2, 蟺/2]\). As \(\tan(x) = \tan(x + k蟺)\) where \(k\) is an integer. Therefore find \(k\) such that the new argument inside \(\tan\) function falls within the interval. In this case, the closest \(\frac{\pi}{2}\) multiple to \(-\frac{25 \pi}{7}\) is \(-\frac{14 \pi}{2} = -7 \pi\), hence, \(k= -7\). We get \(\tan(-\frac{25 \pi}{7} + (-7)\pi)\). By simplifying, it results in \(\tan(\frac{24 \pi}{7})\).
02

Compute the tangent

By reducing the previous result between \(-蟺/2\) and \(蟺/2\), we can simplify it to \(\tan(-\frac{3 \pi}{7})\) because tangent is periodic with a period of \(蟺\). Using a calculator, the tangent of this angle can be computed.
03

Round the answer

Lastly, the value obtained from the previous step must be rounded to four decimal places to obtain the final answer.

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Most popular questions from this chapter

In Exercises 69 and 70, use a graphing utility to graph \(f\) and \(g\) in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. $$ f(x)=\sin (\arctan 2 x), \quad g(x)=\frac{2 x}{\sqrt{1+4 x^{2}}} $$

$$ \text { In Exercises 77-82, sketch a graph of the function. } $$ $$ f(x)=\frac{\pi}{2}+\arctan x $$

Write an algebraic expression that is equivalent to the expression. $$ \sec [\arcsin (x-1)] $$

Describe the behavior of \(f(\theta)=\sec \theta\) at the zeros of \(g(\theta)=\cos \theta\). Explain your reasoning.

In Exercises 1-16, evaluate the expression without using a calculator. $$ \tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right) $$
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