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Chapter 4: Problem 22

Evaluate (if possible) the sine, cosine, and tangent at the real number. $$t=-2 \pi$$

Short Answer

Expert verified
\(\sin (-2\pi) = 0\), \(\cos (-2\pi) = 1\) and \(\tan (-2\pi) = 0\)

Step by step solution

01

- Evaluate Sine at Point

The first task is to compute the sine of \(t\). The sine function has a period of \(2\pi\). Therefore, sine of any real number \(x\) and \(x+2\pi k\) where \(k\) is an integer are equal. Thus \(sin(-2\pi) = sin(0) = 0\).
02

- Evaluate Cosine at Point

Next, compute the cosine of \(t\). The cosine function also has a period of \(2\pi \). Hence, for any real number \(x\) and \(x+2\pi k\) where \(k\) is an integer, their cosine values are equal. Thus \(cos(-2\pi) = cos(0) = 1\).
03

- Evaluate Tangent at Point

Finally, compute the tangent of \(t\). Since tangent function is the ratio of sine to cosine, and here we have \(sin(t) = 0\) and \(cos(t) = 1\), we get \(tan(-2\pi) = sin(-2\pi)/cos(-2\pi) = 0/1 = 0 \).

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