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Chapter 4: Problem 29

Evaluate (if possible) the six trigonometric functions at the real number. $$t=-\pi / 2$$

Short Answer

Expert verified
The trigonometric functions evaluated at \(-\pi / 2\) are as follows: \(\sin(-\pi / 2) = -1\), \(\cos(-\pi / 2) = 0\), \(\tan(-\pi / 2)\) = undefined, \(\csc(-\pi / 2) = -1\), \(\sec(-\pi / 2)\) = undefined, \(\cot(-\pi / 2)\) = undefined.

Step by step solution

01

Calculate Sine

We know that \(\sin(x)\) reaches its minimum value of -1 at \(x = -\pi/2\). so \(\sin(-\pi / 2) = -1\).
02

Calculate Cosine

\(\cos(x)\) is zero at \(x = -\pi/2\), so \(\cos(-\pi / 2) = 0\).
03

Calculate Tan

Since \(\tan(x) = \sin(x)/\cos(x)\), and we previously found that \(\sin(-\pi / 2) = -1\) and \(\cos(-\pi / 2) = 0\), we see that the tangent at this point is undefined, because the denominator is zero.
04

Calculate Cosecant

The cosecant is the reciprocal of the sine, so since \(\sin(-\pi / 2) = -1\), \(\csc(-\pi / 2) = 1/(-1) = -1\).
05

Calculate Secant

The secant is the reciprocal of the cosine. Since \(\cos(-\pi / 2) = 0\), the secant function is not defined at \(-\pi / 2\).
06

Calculate Cotangent

The cotangent is the reciprocal of the tangent. Since the tangent is not defined at \(-\pi / 2\), the cotangent is also not defined at \(-\pi / 2\).

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