WEBVTT
00:00:00.980 --> 00:00:05.310
The given tables show some values of π of π₯, π of π₯, and β of π₯.
00:00:05.790 --> 00:00:09.650
Which function corresponds to the function sin inverse of π₯?
00:00:10.780 --> 00:00:19.300
So as the question says, we have three tables and they represent values of π of π₯, π of π₯, and β of π₯, respectively.
00:00:20.330 --> 00:00:29.490
One of these functions π of π₯, π of π₯, or β of π₯ is sin inverse of π₯, and weβre going to go through each option in turn to work out which one it is.
00:00:30.370 --> 00:00:36.700
So first, we look at the table corresponding to π of π₯ and we ask, βis π of π₯ equal to sin inverse of π₯?β
00:00:37.760 --> 00:00:46.660
Well, if it is, then sin of π of π₯ is equal to sin of sin inverse of π₯, and sin of sin inverse of π₯ is equal to π₯.
00:00:46.840 --> 00:00:50.450
So really, weβre asking if sin of π of π₯ is equal to π₯.
00:00:51.170 --> 00:00:54.230
And we can test this using the values in the table.
00:00:55.380 --> 00:01:01.960
So we look at the first pair of values, where π₯ is negative root three by two and π of π₯ is negative π by three.
00:01:02.710 --> 00:01:07.040
And we ask for this pair of values if sin π of π₯ is equal to π₯.
00:01:07.610 --> 00:01:14.520
So therefore, sin negative π by three has to be equal to negative root three by two.
00:01:15.420 --> 00:01:22.070
And either by knowing something about special angles or else using our calculator, we can see that this is in fact true.
00:01:22.860 --> 00:01:26.500
But of course, we need it to be true for the other values of π₯ and π of π₯.
00:01:28.210 --> 00:01:34.370
And so we look at the second pair of values, π₯ is negative a half and π of π₯ is negative π by six.
00:01:34.760 --> 00:01:40.880
And so we have to ask, βis sin of negative π by six equal to negative a half?β
00:01:41.650 --> 00:01:42.720
And of course, it is.
00:01:42.720 --> 00:01:46.100
So itβs looking good for π of π₯, but we need to check all the values.
00:01:46.100 --> 00:01:54.420
So moving on, so looking at the next pair, we ask if a sin of zero is equal to zero, and of course it is.
00:01:55.490 --> 00:02:03.380
So moving on to the next pair of values, we ask if sin of π by four is equal to a half, and of course it isnβt.
00:02:03.380 --> 00:02:07.190
Sin π by four is equal to root two over two and not a half.
00:02:07.970 --> 00:02:12.720
And this means that for this pair of values, sin of π of π₯ is not equal to π₯.
00:02:13.590 --> 00:02:16.980
And so π of π₯ is not sin inverse of π₯.
00:02:18.270 --> 00:02:22.940
And so π of π₯ is not our answer, and we move on to considering π of π₯.
00:02:23.940 --> 00:02:28.430
So itβs exactly the same process but for π of π₯ instead of π of π₯.
00:02:29.380 --> 00:02:37.510
To find out if π of π₯ is equal to sin inverse of π₯, we ask if sin of π of π₯ is equal to π₯, for all the values in our table.
00:02:38.410 --> 00:02:46.590
So considering the first pair of values, we ask if sin of negative π by six is equal to negative root three over two.
00:02:47.300 --> 00:02:53.120
And the answer here is no; sin of negative π by six is actually equal to negative a half.
00:02:53.800 --> 00:02:55.700
Weβve fallen at the first hurdle.
00:02:55.700 --> 00:02:58.650
Sin of π of π₯ is not equal to π₯, in general.
00:02:59.050 --> 00:03:01.260
For this pair of values, it doesnβt hold, certainly.
00:03:02.400 --> 00:03:05.370
And so π of π₯ is not sin inverse of π₯.
00:03:06.260 --> 00:03:11.210
Now that weβve eliminated both π of π₯ and π of π₯, weβve only got one option left, β of π₯.
00:03:12.110 --> 00:03:18.850
Iβm not going to spend too much time on this but letβs just check that sin of β of π₯ is equal to π₯, for all the values in the table.
00:03:19.700 --> 00:03:37.140
It turns out that sin of negative π by two is negative one, sin of negative π by six is equal to negative a half, sin of zero is zero, sin of π by four is root two over two, and sin of π by three is root three over two.
00:03:38.110 --> 00:03:41.720
For all the values in our table, sin of β of π₯ is equal to π₯.
00:03:41.920 --> 00:03:45.810
And so we conclude that β of π₯ is sin inverse of π₯.