Tables of Values for Trigonometric Functions in .NET framework

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4.3 Tables of Values for Trigonometric Functions
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In this chapter, Table 1 or Table 2 will be used to find values of trigonometric functions whenever a manual solution is used. If the angle contains a number of degrees only or a number of degrees and a multiple of 10 , the value of the function is read directly from the table.
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CHAPTER 4 Solution of Right Triangles
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EXAMPLE 4.1 Find sin 24 40 .
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Opposite 24 40 (< 45 ) in the left-hand column read the entry 0.4173 in the column labeled sin A at the top.
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EXAMPLE 4.2 Find cos 72 .
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Opposite 72 (>45 ) in the right-hand column read the entry 0.3090 in the column labeled cos A at the bottom.
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EXAMPLE 4.3 (a) tan 55 20
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1.4460. Read up the page since 55 20 > 45 . 1.1171. Read down the page since 41 50 < 45
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(b) cot 41 50
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If the number of minutes in the given angle is not a multiple of 10, as in 24 43 , interpolate between the values of the functions of the two nearest angles (24 40 and 24 50 ) using the method of proportional parts.
EXAMPLE 4.4 Find sin 24 43 .
We find
sin 24 40 sin 24 50 Difference for 10
0.4173 0.4200 0.0027 tabular difference
Correction
difference for 3
0.3(0.0027)
0.00081 or 0.0008 when rounded off to four decimal places.
As the angle increases, the sine of the angle increases; thus, sin 24 43 0.4173 0.0008 0.4181
If a five-place table is available, the value 0.41813 can be read directly from the table and then rounded off to 0.4181.
EXAMPLE 4.5 Find cos 64 26 .
We find
cos 64 20 cos 64 30 Tabular difference
0.4331 0.4305 0.0026
Correction
0.6(0.0026)
0.00156 or 0.0016 to four decimal places.
As the angle increases, the cosine of the angle decreases. Thus cos 64 26 0.4331 0.0016 0.4315
To save time, we should proceed as follows in Example 4.4: (a) Locate sin 24 40 0.4173. For the moment, disregard the decimal point and use only the sequence 4173.
(b) Find (mentally) the tabular difference 27, that is, the difference between the sequence 4173 corresponding to 24 40 and the sequence 4200 corresponding to 24 50 . (c) Find 0.3(27) 8.1 and round off to the nearest integer. This is the correction.
(d) Add (since sine) the correction to 4173 and obtain 4181. Then sin 24 43 0.4181
When, as in Example 4.4, we interpolate from the smaller angle to the larger, (1) the correction is added in finding the sine, tangent, and secant; and (2) the correction is subtracted in finding the cosine, cotangent, and cosecant.
CHAPTER 4 Solution of Right Triangles
EXAMPLE 4.6 Find cos 27.23 .
We find
cos 27.20 cos 27.30 Tabular difference
0.8894 0.8886 0.0008
Correction
0.3(0.0008)
0.00024 or 0.0002 to four decimal places.
As the angle increases, the cosine decreases, and thus cos 27.23 0.8894 0.0002 0.8892
EXAMPLE 4.7 Find sec 57.08 .
We find
sec 57.00 sec 57.10 Tabular difference
1.8361 1.8410 0.0049
Correction
0.8(0.0049)
0.00392 or 0.0039 to four decimal places.
As the angle increases, the secant increases, and thus sec 57.08 1.8361 0.0039 1.8400 (See Probs. 4.1 and 4.2.)
4.4 Using Tables to Find an Angle Given a Function Value
The process is a reversal of that given above.
EXAMPLE 4.8 Reading directly from Table 1, we find
EXAMPLE 4.9 Find A, given sin A
sin 17
tan 70 10
0.4234. (Use Table 1.)
The given value is not an entry in the table. We find, however, 0.4226 0.4253 0.0027 Correction
0.0008 0.0027
sin 25 0 sin 25 10 tabular difference
8 27
0.4226 0.4234 0.0008
sin 25 0 sin A partial difference
(10r)
(10r)
3r, to the nearest minute. 3 25 3 A.
Adding (since sine) the correction, we have 25 0
EXAMPLE 4.10 Find A, given cot A
0.6345. (Use Table 1.) cot 57 40 cot 57 30 tabular difference 0.6330 0.6345 0.0015 cot 57 40 cot A partial difference
We find
0.6330 0.6371 0.0041
CHAPTER 4 Solution of Right Triangles
0.0015 0.0041 15 41 (10r)
Correction
(10r)
4r, to the nearest minute. 4 57 36 A.
Subtracting (since cotangent) the correction, we have 57 40 To save time, we should proceed as follows in Example 4.9: (a) Locate the next smaller entry: 0.4226 (b) Find the tabular difference, 27.
sin 25 0 . For the moment, use only the sequence 4226.
(c) Find the partial difference, 8, between 4226 and the given sequence 4234. (d) Find 27 (10r)
3r and add to 25 0 . 0.4234. (Use Table 2.)
EXAMPLE 4.11 Find A, given sin A
The given value is not an entry in the table. We find 0.4226 0.4242 0.0016 Correction
0.0008 0.0016
sin 25.00 sin 25.10 tabular difference
0.4226 0.4234 0.0008
sin 25.00 sin A partial difference
(0.1)
0.05, to the nearest hundredth. 25.00 0.05 25.05 .
Adding (since sine) the correction, we have A
EXAMPLE 4.12 Find A, given cot A
0.6345. (Use Table 2.) cot 57.60 cot 57.50 tabular difference 0.6322 0.6345 0.0023 cot 57.60 cot A partial difference
We find
0.6322 0.6346 0.0024
0.0023 0.0024
Correction
(0.1)
0.10, to the nearest hundredth. 57.60 0.10 57.50 . (See Prob. 4.4.)
Subtracting (since cotangent) the correction, we have A
4.5 Calculator Values of Trigonometric Functions
Calculators give values of trigonometric functions based on the number of digits that can be displayed, usually 8, 10, or 12. The number of decimal places shown varies with the size of the number but is usually at least four. When a calculator is used in this book, all trigonometric function values shown will be rounded to six digits unless the value is exact using fewer digits.
EXAMPLE 4.13 Find sin 24 40 .
(a) Put the calculator in degree mode. (b) Press (( ) key, enter 24, press ( ) key, enter 40, press ( ) key, enter 60, press ( )) key, and press (sin) key. (c) sin 24 40 0.417338 rounded to six digits.
EXAMPLE 4.14 Find tan 48 23 . (a) Put the calculator in degree mode.
(b) Press (( ) key, enter 48, press ( ) key, enter 23, press ( ) key, enter 60, press ( )) key, and press (tan) key. (c) tan 48 23 1.12567 rounded to six digits. tan A 48
23 60
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