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Chapter 4: Problem 28

Evaluate (if possible) the six trigonometric functions at the real number. $$t=3 \pi / 2$$

Short Answer

Expert verified
The six trigonometric functions at \(t = \frac{3\pi}{2}\) are as follows: \(sin(t) = -1\), \(cos(t) = 0\), \(tan(t) = undefined\), \(csc(t) = -1\), \(sec(t) = undefined\), \(cot(t) = 0\)

Step by step solution

01

Evaluate sine, cosine and tangent

At \(t = \frac{3\pi}{2}\), the sin, cos, and tan functions are as follows: \[sin(t) = sin(\frac{3\pi}{2}) = -1\], \[cos(t) = cos(\frac{3\pi}{2}) = 0\], \[tan(t) = tan(\frac{3\pi}{2}) = undefined\] because tangent is equal to \(\frac{sin(t)}{cos(t)}\) and division by zero is undefined.
02

Evaluate cosecant, secant, and cotangent

The even less common trigonometric functions - cosecant, secant and cotangent - are simply the reciprocal of sine, cosine, and tangent, respectively. At \(t = \frac{3\pi}{2}\), we can calculate them as: \[csc(t) = csc(\frac{3\pi}{2}) = -1\], \[sec(t) = sec(\frac{3\pi}{2}) = undefined\] because the secant is equal to \(\frac{1}{cos(t)}\) and division by zero is undefined, \[cot(t) = cot(\frac{3\pi}{2}) = 0\] because the cotangent is equal to \(\frac{cos(t)}{sin(t)}\) and \(cos(\frac{3\pi}{2})\) is 0.

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