|
TABEL SINUS,
COSINUS, DAN TANGEN
|
||||||||||||||||||||
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
|
0 o
1 o
2 o
3 o
4 o
5 o
6 o
7 o
8 o
9 o
10 o
11 o
12 o
13 o
14 o
15 o
16 o
17 o
18 o
19 o
20 o
21 o
22 o
23 o
24 o
25 o
26 o
27 o
28 o
29 o
30 o
31 o
32 o
33 o
34 o
35 o
36 o
37 o
38 o
39 o
40 o
41 o
42 o
43 o
44 o
45 o
|
0.000
0.017
0.035
0.052
0.070
0.087
0.105
0.122
0.139
0.156
0.174
0.191
0.208
0.225
0.242
0.259
0.276
0.292
0.309
0.326
0.342
0.358
0.375
0.391
0.407
0.423
0.438
0.454
0.469
0.485
0.500
0.515
0.530
0.545
0.559
0.574
0.588
0.602
0.616
0.629
0.643
0.656
0.669
0.682
0.695
0.707
|
1.000
1.000
0.999
0.999
0.998
0.996
0.995
0.993
0.990
0.988
0.985
0.982
0.978
0.974
0.970
0.966
0.961
0.956
0.951
0.946
0.940
0.934
0.927
0.921
0.914
0.906
0.899
0.891
0.883
0.875
0.866
0.857
0.848
0.839
0.829
0.819
0.809
0.799
0.788
0.777
0.766
0.755
0.743
0.731
0.719
0.707
|
0.000
0.017
0.035
0.052
0.070
0.087
0.105
0.123
0.141
0.158
0.176
0.194
0.213
0.231
0.249
0.268
0.287
0.306
0.325
0.344
0.364
0.384
0.404
0.424
0.445
0.466
0.488
0.510
0.532
0.554
0.577
0.601
0.625
0.649
0.675
0.700
0.727
0.754
0.781
0.810
0.839
0.869
0.900
0.933
0.966
1.000
|
45 o
46 o
47 o
48 o
49 o
50 o
51 o
52 o
53 o
54 o
55 o
56 o
57 o
58 o
59 o
60 o
61 o
62 o
63 o
64 o
65 o
66 o
67 o
68 o
69 o
70 o
71 o
72 o
73 o
74 o
75 o
76 o
77 o
78 o
79 o
80 o
81 o
82 o
83 o
84 o
85 o
86 o
87 o
88 o
89 o
90 o
|
0.707
0.719
0.731
0.743
0.755
0.766
0.777
0.788
0.799
0.809
0.819
0.829
0.839
0.848
0.857
0.866
0.875
0.883
0.891
0.899
0.906
0.914
0.921
0.927
0.934
0.940
0.946
0.951
0.956
0.961
0.966
0.970
0.974
0.978
0.982
0.985
0.988
0.990
0.993
0.995
0.996
0.998
0.999
0.999
1.000
1.000
|
0.707
0.695
0.682
0.669
0.656
0.643
0.629
0.616
0.602
0.588
0.574
0.559
0.545
0.530
0.515
0.500
0.485
0.469
0.454
0.438
0.423
0.407
0.391
0.375
0.358
0.342
0.326
0.309
0.292
0.276
0.259
0.242
0.225
0.208
0.191
0.174
0.156
0.139
0.122
0.105
0.087
0.070
0.052
0.035
0.017
0.000
|
1.000
1.036
1.072
1.111
1.150
1.192
1.235
1.280
1.327
1.376
1.428
1.483
1.540
1.600
1.664
1.732
1.804
1.881
1.963
2.050
2.145
2.246
2.356
2.475
2.605
2.747
2.904
3.078
3.271
3.487
3.732
4.011
4.331
4.705
5.145
5.671
6.314
7.115
8.144
9.514
11.430
14.301
19.081
28.636
57.290
#####
|
90 o
91 o
92 o
93 o
94 o
95 o
96 o
97 o
98 o
99 o
100 o
101 o
102 o
103 o
104 o
105 o
106 o
107 o
108 o
109 o
110 o
111 o
112 o
113 o
114 o
115 o
116 o
117 o
118 o
119 o
120 o
121 o
122 o
123 o
124 o
125 o
126 o
127 o
128 o
129 o
130 o
131 o
132 o
133 o
134 o
135 o
|
1.000
1.000
0.999
0.999
0.998
0.996
0.995
0.993
0.990
0.988
0.985
0.982
0.978
0.974
0.970
0.966
0.961
0.956
0.951
0.946
0.940
0.934
0.927
0.921
0.914
0.906
0.899
0.891
0.883
0.875
0.866
0.857
0.848
0.839
0.829
0.819
0.809
0.799
0.788
0.777
0.766
0.755
0.743
0.731
0.719
0.707
|
0.000
-0.017
-0.035
-0.052
-0.070
-0.087
-0.105
-0.122
-0.139
-0.156
-0.174
-0.191
-0.208
-0.225
-0.242
-0.259
-0.276
-0.292
-0.309
-0.326
-0.342
-0.358
-0.375
-0.391
-0.407
-0.423
-0.438
-0.454
-0.469
-0.485
-0.500
-0.515
-0.530
-0.545
-0.559
-0.574
-0.588
-0.602
-0.616
-0.629
-0.643
-0.656
-0.669
-0.682
-0.695
-0.707
|
#####
-57.29
-28.64
-19.08
-14.30
-11.43
-9.514
-8.144
-7.115
-6.314
-5.671
-5.145
-4.705
-4.331
-4.011
-3.732
-3.487
-3.271
-3.078
-2.904
-2.747
-2.605
-2.475
-2.356
-2.246
-2.145
-2.050
-1.963
-1.881
-1.804
-1.732
-1.664
-1.600
-1.540
-1.483
-1.428
-1.376
-1.327
-1.280
-1.235
-1.192
-1.150
-1.111
-1.072
-1.036
-1.000
|
135 o
136 o
137 o
138 o
139 o
140 o
141 o
142 o
143 o
144 o
145 o
146 o
147 o
148 o
149 o
150 o
151 o
152 o
153 o
154 o
155 o
156 o
157 o
158 o
159 o
160 o
161 o
162 o
163 o
164 o
165 o
166 o
167 o
168 o
169 o
170 o
171 o
172 o
173 o
174 o
175 o
176 o
177 o
178 o
179 o
180 o
|
0.707
0.695
0.682
0.669
0.656
0.643
0.629
0.616
0.602
0.588
0.574
0.559
0.545
0.530
0.515
0.500
0.485
0.469
0.454
0.438
0.423
0.407
0.391
0.375
0.358
0.342
0.326
0.309
0.292
0.276
0.259
0.242
0.225
0.208
0.191
0.174
0.156
0.139
0.122
0.105
0.087
0.070
0.052
0.035
0.017
0.000
|
-0.707
-0.719
-0.731
-0.743
-0.755
-0.766
-0.777
-0.788
-0.799
-0.809
-0.819
-0.829
-0.839
-0.848
-0.857
-0.866
-0.875
-0.883
-0.891
-0.899
-0.906
-0.914
-0.921
-0.927
-0.934
-0.940
-0.946
-0.951
-0.956
-0.961
-0.966
-0.970
-0.974
-0.978
-0.982
-0.985
-0.988
-0.990
-0.993
-0.995
-0.996
-0.998
-0.999
-0.999
-1.000
-1.000
|
-1.000
-0.966
-0.933
-0.900
-0.869
-0.839
-0.810
-0.781
-0.754
-0.727
-0.700
-0.675
-0.649
-0.625
-0.601
-0.577
-0.554
-0.532
-0.510
-0.488
-0.466
-0.445
-0.424
-0.404
-0.384
-0.364
-0.344
-0.325
-0.306
-0.287
-0.268
-0.249
-0.231
-0.213
-0.194
-0.176
-0.158
-0.141
-0.123
-0.105
-0.087
-0.070
-0.052
-0.035
-0.017
0.000
|
|||||
|
|
||||||||||||||||||||
|
TABEL SINUS,
COSINUS, DAN TANGEN
|
||||||||||||||||||||
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
α
|
sin α
|
cos α
|
tan α
|
|
|
180 o
181 o
182 o
183 o
184 o
185 o
186 o
187 o
188 o
189 o
190 o
191 o
192 o
193 o
194 o
195 o
196 o
197 o
198 o
199 o
200 o
201 o
202 o
203 o
204 o
205 o
206 o
207 o
208 o
209 o
210 o
211 o
212 o
213 o
214 o
215 o
216 o
217 o
218 o
219 o
220 o
221 o
222 o
223 o
224 o
225 o
|
0.000
-0.017
-0.035
-0.052
-0.070
-0.087
-0.105
-0.122
-0.139
-0.156
-0.174
-0.191
-0.208
-0.225
-0.242
-0.259
-0.276
-0.292
-0.309
-0.326
-0.342
-0.358
-0.375
-0.391
-0.407
-0.423
-0.438
-0.454
-0.469
-0.485
-0.500
-0.515
-0.530
-0.545
-0.559
-0.574
-0.588
-0.602
-0.616
-0.629
-0.643
-0.656
-0.669
-0.682
-0.695
-0.707
|
-1.000
-1.000
-0.999
-0.999
-0.998
-0.996
-0.995
-0.993
-0.990
-0.988
-0.985
-0.982
-0.978
-0.974
-0.970
-0.966
-0.961
-0.956
-0.951
-0.946
-0.940
-0.934
-0.927
-0.921
-0.914
-0.906
-0.899
-0.891
-0.883
-0.875
-0.866
-0.857
-0.848
-0.839
-0.829
-0.819
-0.809
-0.799
-0.788
-0.777
-0.766
-0.755
-0.743
-0.731
-0.719
-0.707
|
0.000
0.017
0.035
0.052
0.070
0.087
0.105
0.123
0.141
0.158
0.176
0.194
0.213
0.231
0.249
0.268
0.287
0.306
0.325
0.344
0.364
0.384
0.404
0.424
0.445
0.466
0.488
0.510
0.532
0.554
0.577
0.601
0.625
0.649
0.675
0.700
0.727
0.754
0.781
0.810
0.839
0.869
0.900
0.933
0.966
1.000
|
225 o
226 o
227 o
228 o
229 o
230 o
231 o
232 o
233 o
234 o
235 o
236 o
237 o
238 o
239 o
240 o
241 o
242 o
243 o
244 o
245 o
246 o
247 o
248 o
249 o
250 o
251 o
252 o
253 o
254 o
255 o
256 o
257 o
258 o
259 o
260 o
261 o
262 o
263 o
264 o
265 o
266 o
267 o
268 o
269 o
270 o
|
-0.707
-0.719
-0.731
-0.743
-0.755
-0.766
-0.777
-0.788
-0.799
-0.809
-0.819
-0.829
-0.839
-0.848
-0.857
-0.866
-0.875
-0.883
-0.891
-0.899
-0.906
-0.914
-0.921
-0.927
-0.934
-0.940
-0.946
-0.951
-0.956
-0.961
-0.966
-0.970
-0.974
-0.978
-0.982
-0.985
-0.988
-0.990
-0.993
-0.995
-0.996
-0.998
-0.999
-0.999
-1.000
-1.000
|
-0.707
-0.695
-0.682
-0.669
-0.656
-0.643
-0.629
-0.616
-0.602
-0.588
-0.574
-0.559
-0.545
-0.530
-0.515
-0.500
-0.485
-0.469
-0.454
-0.438
-0.423
-0.407
-0.391
-0.375
-0.358
-0.342
-0.326
-0.309
-0.292
-0.276
-0.259
-0.242
-0.225
-0.208
-0.191
-0.174
-0.156
-0.139
-0.122
-0.105
-0.087
-0.070
-0.052
-0.035
-0.017
0.000
|
1.000
1.036
1.072
1.111
1.150
1.192
1.235
1.280
1.327
1.376
1.428
1.483
1.540
1.600
1.664
1.732
1.804
1.881
1.963
2.050
2.145
2.246
2.356
2.475
2.605
2.747
2.904
3.078
3.271
3.487
3.732
4.011
4.331
4.705
5.145
5.671
6.314
7.115
8.144
9.514
11.430
14.301
19.081
28.636
57.290
#####
|
270 o
271 o
272 o
273 o
274 o
275 o
276 o
277 o
278 o
279 o
280 o
281 o
282 o
283 o
284 o
285 o
286 o
287 o
288 o
289 o
290 o
291 o
292 o
293 o
294 o
295 o
296 o
297 o
298 o
299 o
300 o
301 o
302 o
303 o
304 o
305 o
306 o
307 o
308 o
309 o
310 o
311 o
312 o
313 o
314 o
315 o
|
-1.000
-1.000
-0.999
-0.999
-0.998
-0.996
-0.995
-0.993
-0.990
-0.988
-0.985
-0.982
-0.978
-0.974
-0.970
-0.966
-0.961
-0.956
-0.951
-0.946
-0.940
-0.934
-0.927
-0.921
-0.914
-0.906
-0.899
-0.891
-0.883
-0.875
-0.866
-0.857
-0.848
-0.839
-0.829
-0.819
-0.809
-0.799
-0.788
-0.777
-0.766
-0.755
-0.743
-0.731
-0.719
-0.707
|
0.000
0.017
0.035
0.052
0.070
0.087
0.105
0.122
0.139
0.156
0.174
0.191
0.208
0.225
0.242
0.259
0.276
0.292
0.309
0.326
0.342
0.358
0.375
0.391
0.407
0.423
0.438
0.454
0.469
0.485
0.500
0.515
0.530
0.545
0.559
0.574
0.588
0.602
0.616
0.629
0.643
0.656
0.669
0.682
0.695
0.707
|
#####
-57.29
-28.64
-19.08
-14.30
-11.43
-9.514
-8.144
-7.115
-6.314
-5.671
-5.145
-4.705
-4.331
-4.011
-3.732
-3.487
-3.271
-3.078
-2.904
-2.747
-2.605
-2.475
-2.356
-2.246
-2.145
-2.050
-1.963
-1.881
-1.804
-1.732
-1.664
-1.600
-1.540
-1.483
-1.428
-1.376
-1.327
-1.280
-1.235
-1.192
-1.150
-1.111
-1.072
-1.036
-1.000
|
315 o
316 o
317 o
318 o
319 o
320 o
321 o
322 o
323 o
324 o
325 o
326 o
327 o
328 o
329 o
330 o
331 o
332 o
333 o
334 o
335 o
336 o
337 o
338 o
339 o
340 o
341 o
342 o
343 o
344 o
345 o
346 o
347 o
348 o
349 o
350 o
351 o
352 o
353 o
354 o
355 o
356 o
357 o
358 o
359 o
360 o
|
-0.707
-0.695
-0.682
-0.669
-0.656
-0.643
-0.629
-0.616
-0.602
-0.588
-0.574
-0.559
-0.545
-0.530
-0.515
-0.500
-0.485
-0.469
-0.454
-0.438
-0.423
-0.407
-0.391
-0.375
-0.358
-0.342
-0.326
-0.309
-0.292
-0.276
-0.259
-0.242
-0.225
-0.208
-0.191
-0.174
-0.156
-0.139
-0.122
-0.105
-0.087
-0.070
-0.052
-0.035
-0.017
0.000
|
0.707
0.719
0.731
0.743
0.755
0.766
0.777
0.788
0.799
0.809
0.819
0.829
0.839
0.848
0.857
0.866
0.875
0.883
0.891
0.899
0.906
0.914
0.921
0.927
0.934
0.940
0.946
0.951
0.956
0.961
0.966
0.970
0.974
0.978
0.982
0.985
0.988
0.990
0.993
0.995
0.996
0.998
0.999
0.999
1.000
1.000
|
-1.000
-0.966
-0.933
-0.900
-0.869
-0.839
-0.810
-0.781
-0.754
-0.727
-0.700
-0.675
-0.649
-0.625
-0.601
-0.577
-0.554
-0.532
-0.510
-0.488
-0.466
-0.445
-0.424
-0.404
-0.384
-0.364
-0.344
-0.325
-0.306
-0.287
-0.268
-0.249
-0.231
-0.213
-0.194
-0.176
-0.158
-0.141
-0.123
-0.105
-0.087
-0.070
-0.052
-0.035
-0.017
0.000
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Teorema Pythagoras
Teorema Pythagoras
Apabila diketahui
panjang dua sisi segitiga siku-siku,
maka panjang sisi yang ketiga dapat dihitung dengan
Teorema Pythagoras.
Contoh Soal:
Hitunglah panjang sisi x yang belum diketahui, pada segitiga siku-siku berikut ini :
B Teorema Pythagoras
:
AB2 = AC2 + BC2
c
atau a
c2 = a2 + b2
(a) x
5 12
Jawab :
Jawab :
x2
= 122 + 52
= 144 + 25
= 169
x = √169 = 13
C
b A
(b)
1
x
43 19
432 = x2
+ 192
x2 = 432 - 192
= 1849 – 361
= 1488
x = √1488 = 38,57460
≈ 38,57 2
Perbandingan Trigonometri
B
Sinus, Cosinus
dan Tangen
Perbandingan Trigonometri
Hubungan perbandingan trigonometri segitiga siku-siku
pada koordinat Cartesius ditunjukkan sbb. :
y
A
sisi depan BC
θ )
samping C
B(x,y)
r
y
θ )
x
sisi depan
sin θ = sisi miring =
sisi samping
cos θ = sisi miring
y r
x
= r
sin θ =
sisi miring
= AB
O x A
sisi depan y
tan θ = =
cos θ =
sisi samping sisi miring
= AC AB
Catatan:
samping x
tan θ =
sisi depan BC
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Hubungan nilai sin θ dan besar sudut θ dapat dilihat pada tabel sinus
4
Perbandingan Trigonometri
Perbandingan Trigonometri
Secan, Cosecan dan Cotangen (sec, csc dan cot)
y
B(x,y)
Perbandingan
trigonometri selain sinus, cosinus
y
dan tangen ada perbandingan yang
lain, yaitu secan, cosecan dan cotangen.
Perbandingan secan, cosecan dan cotangen tersebut adalah
sbb. :
B(x,y)
r
y
Hubungan
antara cos, sin, dan tan dengan sec, csc dan cot adalah kebalikannya.
r
y
θ )
x
O x A
sec θ =
csc θ =
1
cos θ
1
sin θ
|
1
= x
|
1
|
r
1
= 1 x r/x =
= 1 x r/y =
r x
r
y
x
θ )
x
O x A
cos θ =
sin θ =
x r
y
r
sec kebalikan dari cos
csc kebalikan dari sin
sec θ = r x
csc θ = r
y
cot θ =
tan θ
y/x
= 1 x x/y = y
y
5
tan θ = x
cot kebalikan dari tan
x
cot θ = y 6
VEKTOR KOMPONEN DAN
RESULTAN BEBERAPA VEKTOR
DENGAN PENDEKATAN TRIGONOMETRI
Vektor
Y
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Kom
Vektor Komponen
F adalah:
Fx dan Fy.
Besar vektor komponen.
Fx = F.cos α
Fy = F.sin α
7
8
Kwadran II
Sinus Æ positif Y
Kwadran I Semua Æ positif
Hubungan antara besar sudut (α), jari-
Kwadran II
sin Æ positif
Kwadran
I
Y
sem Æ positif
F
Fy Fx = F.cos α
90o
90o
90o 90o
α
X
360o
Sinus Æ sin α=y/r
Cosinus Æ cosα=x/r Tangen Æ tanα=y/x
α
Fx X
Fy = F.sin α
Kwadran III Tangen Æ positif
Kwadran IV Cosinus Æ positif
9
Kwadran
III
tan Æ positif
Kwadran
IV
cos Æ positif
10
Kwadran II
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Kwadran I
sem Æ positif
Kwadran II
sin Æ positif Y
Kwadran I
sem Æ positif
Kwadr
tan Æ positif
Fx = F.cos α
F.(-cos(180o-α)) F.(-cos β)
Fy = F.sin α F.sin(180o-α) F.sin β
Kwadran
IV
cos Æ positif
11
Fx
β
α
F
|
Kwadran III
tan Æ positif
Fx = F.cos α
F.(-cos(α-180o)) F.(-cos β)
X Fy = F.sinα
F.(-sin(α-180o)) F.(-sin β)
Kwadran
IV
cos Æ positif
12
Kwadran II
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Kwadran I
sem Æ positif
Sudut Istimewa
Fx = F.cos α
F.cos(360o-α) F.cos β
Fy = F.sin α
F.(-sin(360o-α))
F.(-sin β) Fy
/2√4
/2√3
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/2√1
/2√0
Kwadran III
tan Æ positif
Kwadran IV
cos Æ positif
13
Untuk besar sudut yang lain (0o s/d 360o) lihat pada tabel !
14
Resultan
Y
Vekt
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Resultan
Beberapa
Y Resultan dari
x
y
Resultan Vektor
Vektor
F2
F1 vektor
F1, F2, F3, dan
F4 adalah FR.
Komponen Fx dan Fy
|
Besar FR.
2 = ΣF
+ ΣF 2
β α FR x2 y
j
2
Besarnya:
2 2
2
θ X
FR = √ ΣFx2+ ΣFy
FR
= Fx +Fy
F Besar ΣF dan ΣF .
|
3
x
|
ΣF =F
+F
y
+F +F
x 1x
2x 3x 4x
ΣFy=F1y+F2y+F3y+F4y
Arah
Y
Resultan
Vektor
F1
|
FR = √ ΣFx2+ ΣFy
Contoh Soal
Y
F1
Perhatikan gambar di samping!
Diketahui:
F =20 N, F =25 N
1 2
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Arah FR :
|
F3=30 N, F4=40 N, F2
sedangkan besar
sudutnya α=60o,
β α β=30o, j=45o, θ=60o.
j θ X
F3
F4
σ = …. derajat
(lihat tabel)
Jadi arah FR =
σ derajat terhadap
sumbu X positif.
j θ X
F3
F4
Hitunglah:
a.Komponen masing- masing vektor.
b.Besar ΣFx. c.Besar ΣFy.
d.Besar F
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e.Arah FR
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