Machinery's Handbook, 31st Edition
Trigonometry Tables 107 their sum is 90 ° ) and are related (see Cofunction Identities: , page 97). For example, sin 10 ° = cos 80 ° and cos 10 ° = sin 80 ° . Expanded trig tables are also available in the AD- DITIONAL material in the Machinery’s Handbook 31 Digital Edition . Angle measures greater than 90 degrees are converted to their reference angle (that is, its angle equivalent) before a trigonometric value can be found. If the angle θ is between 90 and 180, its reference angle is 180 – θ ; if θ is between 180 and 270, then θ – 180 is its reference angle; and if it is between 270 and 360 degrees, 360 – θ is its acute angle equivalent. To determine trigonometric values of functions of angles greater than 90 °, subtract 90, 180, 270, or 360 from the angle to get the reference angle less than 90 ° and use Table 1, Useful Relationships Among Angles , to find the equivalent first-quadrant func - tion and angle to look up in the trig tables.
Radians
1.6 1.5
1.4 1.3
π 3 - -
2 π 3 ---
1.8 1.7
1.9
π 2 --
1.2
2.0
1.1
2.1
1.0
π 4 --
3 π 4 ---
2.2
0.9
2.3
0.8
2.4
0.7
70 100 90 80 110
2.5
π 6 --
5 π 6 ---
0.6
120
60
r e
2.6
50
130
0.5
+ − − − − + − − + + − −
+ + + + + + − + − − + −
sin cos tan cot sec csc sin cos tan cot sec csc
sin cos tan cot sec csc sin cos tan cot sec csc
2.7
(1 to 0) (0 to − 1)
(0 to ) ( to 0) ( to 1) (1 to ) (0 to 1) (1 to 0) ( to 0) (0 to ) ( − 1 to ) ( to 1) ( − 1 to 0) (0 to 1)
40
140
0.4
2.8
30
150
( to 0) (0 to ) (1 to ) ( to − 1)
0.3
2.9
20
160
0.2
3.0
10
170
0.1
3.1 3.2
II I III IV
π
180
350 0 and 360
2 π
6.3
6.2
(0 to ) ( to 0) ( to − 1) ( − 1 to ) (0 to − 1) ( − 1 to 0)
190
3.3
6.1
200
340
3.4
6.0
210
330
3.5
5.9
220
320
3.6
5.8
310
230
3.7
11 π 6 -----
7 π 6 ---
240
300
5.7
250
3.8
260 270 280 290
5.6
3.9
5.5
5 π 4 ---
7 π 4 ---
4.0
5.4
4.1
5.3
4.2
5.2
4.3
4 π 3 ---
5 π 3 ---
5.1
4.4 4.5
5.0
4.6 4.7 4.8 4.9
3 π 2 ---
Fig. 4. Signs of Trigonometric Functions, Fractions of p , and Degree-Radian Conversion Table 1. Useful Relationships Among Angles Angle Function q -q 90 ° ± q 180 ° ± q 270 ° ± q 360 ° ± q sine sin q - sin q +cos q sin q - cos q ± sin q cosine cos q +cos q sin q - cos q ± sin q +cos q tangent tan q - tan q cot q ± tan q cot q ± tan q cotangent cot q - cot q tan q ± cot q tan q ± cot q secant sec q +sec q csc q - sec q ± csc q +sec q cosecant csc q - csc q +sec q csc q - sec q ± csc q Examples: cos (270 ° - q ) = - sin q ; tan (90 ° + q ) = - cot q . Example: Find the cosine of 336 ° 40 ′ . Fig. 4 shows that the cosine of every angle in Quadrant IV (270 ° to 360 ° ) is positive. To find the angle and trig function to use when entering the trig table, subtract 270 from 336 to get cos 336 ° 40 ′ = cos (270 ° + 66 ° 40 ′ ) and then find the intersection of the “cos row” and the 270 ± q column in Table 1. Because cos (270 ± q ) in the fourth quadrant is equal to ± sin q in the first quadrant, find sin 66 ° 40 ′ in the trig table. Therefore, cos 336 ° 40 ′ = sin 66 ° 40 ′ = 0.918216.
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