WEBVTT
00:00:01.040 --> 00:00:09.990
Find the value of the measure of angle 𝐸 given that angle 𝐸 is an acute angle and cos of 𝐸 is 0.5201.
00:00:10.570 --> 00:00:12.940
Give your answer to the nearest second.
00:00:13.250 --> 00:00:15.300
So we have a trigonometric equation here.
00:00:15.300 --> 00:00:19.200
cos of 𝐸 is equal to 0.5201.
00:00:19.450 --> 00:00:25.490
We’re going to need to solve this equation for 𝐸 to help us find the value of the measure of angle 𝐸.
00:00:25.870 --> 00:00:31.330
And since the question tells us to give our answer to the nearest second, that tells us two things.
00:00:31.720 --> 00:00:38.200
Firstly, we know we’re going to be working in degrees, minutes, and seconds, rather than, of course, radian measure.
00:00:38.500 --> 00:00:45.780
Secondly, we know that since we’re going to be rounding our answer, that implies that we’re going to be using a calculator to work this one out.
00:00:46.050 --> 00:00:49.500
So how do we solve a trigonometric equation of this form?
00:00:50.010 --> 00:00:55.280
A really common mistake here is to think that, to solve this equation, we divide both sides by cos.
00:00:55.580 --> 00:01:05.170
But in fact, cos itself is an operation and the opposite operation, the inverse to finding the cos of something, is to find the inverse cos.
00:01:05.470 --> 00:01:08.770
And so we find the inverse cos of both sides of our equation.
00:01:08.990 --> 00:01:18.140
The inverse cos of cos of 𝐸 is just 𝐸, and so 𝐸 must be equal to the inverse cos of 0.5201.
00:01:18.600 --> 00:01:20.630
Let’s type this into our calculator.
00:01:21.050 --> 00:01:28.570
The inverse cos of 0.5201 is 58.66104 and so on degrees.
00:01:28.860 --> 00:01:32.690
Now, of course, we need to convert this to degrees, minutes, and seconds.
00:01:32.940 --> 00:01:35.720
And so we could hit the relevant button on our calculator.
00:01:35.930 --> 00:01:37.680
It looks a little something like this.
00:01:37.900 --> 00:01:42.140
Should we not have that functionality, though, we can work this out by hand.
00:01:42.260 --> 00:01:46.320
We’re going to subtract the integer part of our number; we’re going to subtract 58.
00:01:46.550 --> 00:01:49.550
And so we get 0.6610 and so on.
00:01:49.780 --> 00:01:55.270
We know that when we’re working with degrees, minutes, and seconds we’re actually working in base 60.
00:01:55.270 --> 00:01:58.770
So we take this decimal part and we multiply it by 60.
00:01:59.100 --> 00:02:02.680
That gives us 39.6624 and so on.
00:02:03.080 --> 00:02:05.510
So that’s the minutes part of our answer.
00:02:05.860 --> 00:02:08.840
To find the seconds part, we repeat the process once again.
00:02:09.190 --> 00:02:15.920
We take away the integer part, leaving us with 0.66 and so on, and then we multiply that by 60.
00:02:16.130 --> 00:02:21.560
That’s 39.74, which is roughly 40 correct to the nearest second.
00:02:22.420 --> 00:02:29.190
So the degrees part of our answer is 58, the minutes part is 39, and the seconds part is 40.
00:02:29.410 --> 00:02:32.080
This is indeed an acute angle as required.
00:02:32.080 --> 00:02:38.310
And so we find the measure of angle 𝐸 is 58 degrees, 39 minutes, and 40 seconds.