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Chapter 8: Problem 38

Determine the value of \(\cos t\) when \(t=5 \pi,-2 \pi, 17 \pi / 2\), \(-13 \pi / 2\)

Short Answer

Expert verified
For \(t = 5 \pi, \text{cos}(t) = -1\), for \(-2 \pi, \text{cos}(t) = 1\), for \(17 \pi / 2, \text{cos}(t) = 0\), and for \(-13 \pi / 2, \text{cos}(t) = 0\).

Step by step solution

01

Understand the problem

We need to determine the value of \(\text{cos}(t)\) for the given values of \(t\).
02

Determine the value for \(t = 5 \pi \)

The cosine function has a period of \(2 \pi\). Therefore, we can simplify \(t = 5 \pi \) by reducing it within one period: \(5 \pi \text{ mod } 2 \pi = \pi\). \(\text{cos}(5 \pi) = \text{cos}(\pi) = -1 \).
03

Determine the value for \(t = -2 \pi \)

For \(t = -2 \pi \), since \(\text{cos}\) is a periodic function with period \(2 \pi\), \(\text{cos}(-2 \pi) = \text{cos}(0) = 1 \).
04

Determine the value for \(t = 17 \pi / 2 \)

First simplify \(t = 17 \pi / 2 \) within one period. \(17 \pi / 2\text{ mod } 2 \pi = \pi / 2\). Thus, \(\text{cos}(17 \pi / 2) = \text{cos}( \pi / 2) = 0 \).
05

Determine the value for \(t = -13 \pi / 2 \)

Simplify \(t = -13 \pi / 2 \) within one period. \(-13 \pi / 2\text{ mod } 2 \pi = \- \pi / 2\). As \(\text{cos}(- \pi / 2) = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, \(\text{cos}(t)\), is a fundamental trigonometric function that helps link the angles of a triangle to the lengths of its sides. It works with right-angled triangles where the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. The cosine function can take any real number as an input, and it outputs a value between -1 and 1.

The function is defined as: \(\text{cos}(t) = \frac{\text{Adj. Side}}{\text{Hypotenuse}}\).
By understanding this basic definition, we can better solve trigonometric problems involving angles and triangle sides.
Periodic Functions
Periodic functions repeat their values at regular intervals. The cosine function is a classic example of such functions. It repeats every \(2 \pi\) radians, which means \(\text{cos}(t + 2 \pi) = \text{cos}(t)\) for any angle \(t\).

The period of the cosine function, \(2 \pi\) is essential as it helps simplify calculations. By knowing the period, we can adjust any angle to a simpler form that falls within one period. This way, we can handle very large or very small angles effectively.
Modulo Operation
The modulo operation is a mathematical tool that finds the remainder after the division of one number by another. In trigonometry, it helps us reduce angles to simpler equivalents.

For example, \(5\pi\) mod \(2\pi\) = \(\pi\) because dividing \(5\pi\) by \(2\pi\) leaves a remainder of \(\pi\). Using this operation, we can condense angles to more manageable forms, which is often necessary for trigonometric evaluations.
Angle Reduction
Angle reduction is the process of transforming angles into an equivalent angle within a specific range, often from \(0\) to \(2 \pi\). It uses the periodic nature of trigonometric functions. For instance, reducing \(17\pi/2\) within one period gives us \(\pi/2\). We achieve this by applying the modulo operation: \(17\pi/2\) mod \(2\pi\) = \(\pi/2\)

Reduction simplifies trigonometric evaluations as we can work with familiar and fundamental angles like \(\pi\), \(\pi/2\), and 0.

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Most popular questions from this chapter

Differentiate (with respect to \(t\) or \(x\) ): \(y=\cos ^{3} t\)

In any given locality, the length of daylight varies during the year. In Des Moines, Iowa, the number of minutes of daylight in a day \(t\) days after the beginning of a year is given approximately by the formula $$ D=720+200 \sin \left[\frac{2 \pi}{365}(t-79.5)\right], \quad 0 \leq t \leq 365 . $$ (Source: School Science and Mathematics.) (a) Graph the function in the window \([0,365]\) by \([-100,940]\) (b) How many minutes of daylight are there on February 14, that is, when \(t=45 ?\) (c) Use the fact that the value of the sine function ranges from \(-1\) to 1 to find the shortest and longest amounts of daylight during the year. (d) Use the TRACE feature or the MINIMUM command to estimate the day with the shortest amount of daylight. Find the exact day algebraically by using the fact that \(\sin (3 \pi / 2)=-1\) (e) Use the TRACE feature or the MAXIMUM command to estimate the day with the longest amount of daylight. Find the exact day algebraically by using the fact that \(\sin (\pi / 2)=1\). (f) Find the two days during which the amount of daylight equals the amount of darkness. (These days are called equinoxes.) [Note: Answer this question both graphically and algebraically.]

Find the area under the curve \(y=\cos t\) from \(t=0\) to \(t=\frac{\pi}{2}\).

Find \(t\) such that \(0 \leq t \leq \pi\) and \(t\) satisfies the stated condition. \(\cos t=\cos (5 \pi / 4)\)

Find the slope of the line tangent to the graph of \(y=\cos 3 x\) at \(x=13 \pi / 6\).
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