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Chapter 4: Problem 30

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

Short Answer

Expert verified
The six trigonometric functions evaluated at \(t=-\pi\) are \(sin(-\pi) = 0\), \(cos(-\pi) = -1\), \(tan(-\pi) = 0\), \(csc(-\pi)\) is not defined, \(sec(-\pi) = -1\), and \(cot(-\pi)\) is not defined.

Step by step solution

01

Determine the value of sine

The sine (abbreviated 'sin') of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The sin function varies between -1 and 1, and sin of \(t=-\pi\) is 0. Therefore, \(sin(-\pi) = 0\).
02

Determine the value of cosine

The cosine (abbreviated 'cos') of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Cosine function also varies between -1 and 1. The cos of \(t=-\pi\) is -1. Therefore, \(cos(-\pi) = -1\).
03

Determine the value of tangent

The tangent (abbreviated 'tan') of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. It can also be expressed as the ratio of sine to cosine. As sine is 0 and cosine is -1, the tangent of \(t=-\pi\) is 0. Therefore, \(tan(-\pi) = 0\).
04

Determine the value of cosecant

The cosecant (abbreviated 'csc') of an angle in a right triangle is defined as the reciprocal of the sine of the angle. However, the sine value of \(t=-\pi\) is 0 and the reciprocal of 0 is undefined. Therefore, \(csc(-\pi)\) is not defined.
05

Determine the value of secant

The secant (abbreviated 'sec') of an angle in a right triangle is defined as the reciprocal of the cosine of the angle. The cosine value of \(t=-\pi\) is -1 and its reciprocal is also -1, \(sec(-\pi) = -1\).
06

Determine the value of cotangent

The cotangent (abbreviated 'cot') of an angle in a right triangle is defined as the reciprocal of the tangent of the angle. However, the tangent value of \(t=-\pi\) is 0 and the reciprocal of 0 is undefined. Therefore, \(cot(-\pi)\) is not defined.

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Most popular questions from this chapter

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{2} \cos 20 \pi t$$

Find the distance between Dallas, Texas, whose latitude is \(32^{\circ} 47^{\prime} 39^{\prime \prime} \mathrm{N}\) and Omaha, Nebraska, whose latitude is \(41^{\circ} 15^{\prime} 50^{\prime \prime} \mathrm{N}\) Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (Omaha is due north of Dallas).

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse trigonometric function.

The normal monthly high temperatures \(H\) (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $$H(t)=56.94-20.86 \cos \left(\frac{\pi t}{6}\right)-11.58 \sin \left(\frac{\pi t}{6}\right)$$ and the normal monthly low temperatures \(L\) are approximated by $$L(t)=41.80-17.13 \cos \left(\frac{\pi t}{6}\right)-13.39 \sin \left(\frac{\pi t}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January (see figure). (Source: National Climatic Data Center (GRAPH CANNOT COPY). (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June \(21,\) but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Use a graphing utility to graph the function. $$f(x)=\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{\pi}\right)$$
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