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Chapter 1: Problem 17

\(\cdot\) An angle is given, to one significant figure, as \(4^{\circ},\) meaning that its value is between \(3.5^{\circ}\) and \(4.5^{\circ} .\) Find the corresponding range of possible values of \((a)\) the cosine of the angle, (b) the sine of the angle, and (c) the tangent of the angle.

Short Answer

Expert verified
Cosine: [0.996, 0.998]; Sine: [0.061, 0.078]; Tangent: [0.061, 0.079].

Step by step solution

01

Understand the Range of the Angle

The angle is given as \(4^{\circ}\) and is specified to be between \(3.5^{\circ}\) and \(4.5^{\circ}\). Therefore, our task is to find the cosine, sine, and tangent for the range of angles from \(3.5^{\circ}\) to \(4.5^{\circ}\).
02

Calculate the Cosine Values

To find the cosine values for the angle range, compute the cosine for both the minimum and maximum angles: \(\cos(3.5^{\circ})\) and \(\cos(4.5^{\circ})\). Use a calculator to find these precise values:\[\cos(3.5^{\circ}) \approx 0.998 \quad \text{and} \quad \cos(4.5^{\circ}) \approx 0.996\]Thus, the range of cosine values is approximately \([0.996, 0.998]\).
03

Calculate the Sine Values

Next, find the sine for the angle range. Calculate \(\sin(3.5^{\circ})\) and \(\sin(4.5^{\circ})\):\[\sin(3.5^{\circ}) \approx 0.061 \quad \text{and} \quad \sin(4.5^{\circ}) \approx 0.078\]So, the range of sine values is approximately \([0.061, 0.078]\).
04

Calculate the Tangent Values

Finally, compute the tangent for both the lower and upper bounds:\[\tan(3.5^{\circ}) \approx 0.061 \quad \text{and} \quad \tan(4.5^{\circ}) \approx 0.079\]Thus, the range of tangent values is approximately \([0.061, 0.079]\).

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Key Concepts

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

Understanding Cosine
The cosine of an angle in a right triangle relates the angle to the ratio of the adjacent side over the hypotenuse.
Think of it simply as a way to measure the "horizontal stretch."
When we talk about the `cosine` of small angles, like those near 4 degrees, the values are close to 1 because the triangle is almost like a flat line.
  • For our specific angle range between 3.5掳 and 4.5掳, the cosine doesn't vary much.
  • As calculated, \[\cos(3.5^{\circ}) \approx 0.998\] li> and \[\cos(4.5^{\circ}) \approx 0.996\].
Thus, our range is narrow, highlighting how cosine is always close to 1 for tiny angles.
Cosine is a fundamental concept because it's often used to resolve forces in physics and calculations in engineering.
Understanding Sine
The sine function measures the "vertical stretch" or the ratio of the opposite side to the hypotenuse in a right triangle.
Similar to cosine, for small angles, sine values are also tiny, but they increase slowly with the angle.
  • In our case, \[\sin(3.5^{\circ}) \approx 0.061\]
  • and \[\sin(4.5^{\circ}) \approx 0.078\].
The small range between these values shows that even minimal angle changes can slightly increase sine values.
Sine is crucial in scenarios where we need to calculate heights or when dealing with oscillatory motions such as waves.
Understanding Tangent
The tangent of an angle provides the ratio of the sine and cosine, or the opposite side divided by the adjacent side in a triangle.
The `tan` function tells us how steep a line is from the viewpoint of the angle.
  • For very small angles like 3.5掳 to 4.5掳, tangent values are also minimal.
  • We calculated \[\tan(3.5^{\circ}) \approx 0.061\]
  • and \[\tan(4.5^{\circ}) \approx 0.079\].
Here, both values indicate a gentle slope, as expected from these small angles.
`Tangent` has practical applications in physics for calculating speed or angles of elevation and depression in navigation.

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Most popular questions from this chapter

\(\cdot\) Express each of the following numbers to three, five, and eight significant figures: (a) \(\pi=3.141592654 \ldots,\) (b) \(e=\) \(2.718281828 \ldots,(\mathrm{c}) \sqrt{13}=3.605551275 \ldots\)

. Breathing oxygen. The density of air under standard laboratory conditions is \(1.29 \mathrm{kg} / \mathrm{m}^{3},\) and about 20\(\%\) of that air consists of oxygen. Typically, people breathe about \(\frac{1}{2} \mathrm{L}\) of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?

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\(\cdot\) On a single diagram, carefully sketch each force vector to scale and identify its magnitude and direction on your drawing: (a) 60 lb at \(25^{\circ}\) east of north. (b) 40 lb at \(\pi / 3\) south of west. (c) 100 lb at \(40^{\circ}\) north of west. (d) 50 lb at \(\pi / 6\) east of south.

A rocket fires two engines simultaneously. One produces a thrust of 725 \(\mathrm{N}\) directly forward, while the other gives a \(513-\mathrm{N}\) thrust at \(32.4^{\circ}\) above the forward, while the other gives a \(513-\mathrm{N}\) and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.
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