WEBVTT
00:00:00.720 --> 00:00:04.440
Part a) Calculate the length 𝑥.
00:00:06.480 --> 00:00:13.880
Looking at the diagram we’ve been given, we can see that we have a right-angled triangle in which 𝑥 represents the length of one of the sides.
00:00:15.160 --> 00:00:23.080
As we want to calculate the length of a side in a right-angled triangle, there are two approaches that spring to mind: Pythagoras theorem or trigonometry.
00:00:23.760 --> 00:00:29.440
However, if we wanted to use Pythagoras theorem, we need to know both of the other two side lengths.
00:00:29.720 --> 00:00:33.240
And from the diagram, we can see that we’re only given one side length.
00:00:34.560 --> 00:00:37.880
We’re also given one of the other angles apart from the right angle.
00:00:38.160 --> 00:00:41.880
So this tells us that we’re going to need to use trigonometry for this question.
00:00:43.200 --> 00:00:48.360
The first step in any problem involving trigonometry is to label the three sides of the triangle.
00:00:49.760 --> 00:00:55.800
First, we have the hypotenuse, H, the longest side of the triangle, which is always directly opposite the right angle.
00:00:57.000 --> 00:01:01.480
We then have the opposite side, O, which is opposite the other angle that we’ve been given.
00:01:01.840 --> 00:01:03.760
In this case, that’s the angle of 50 degrees.
00:01:05.120 --> 00:01:10.200
And finally, we have the adjacent side, A, which is between the right angle and the angle we’ve been given.
00:01:11.640 --> 00:01:18.560
We can then use the memory aid SOHCAHTOA to help us decide which of the three trigonometric ratios we need in this question.
00:01:19.240 --> 00:01:26.040
Here, S, C, and T stand for sin, cos, and tan and O, A, and H stand for opposite, adjacent, and hypotenuse.
00:01:27.440 --> 00:01:31.720
The side we know is the adjacent, and the side we want to calculate is the hypotenuse.
00:01:32.040 --> 00:01:37.560
So the pair of sides that are involved in the ratio are A and H, which is the cos ratio.
00:01:38.080 --> 00:01:39.440
Let’s recall its definition.
00:01:40.880 --> 00:01:47.200
Cos or cosine of an angle 𝜃 is equal to the length of the adjacent side divided by the length of the hypotenuse.
00:01:48.800 --> 00:01:51.600
We can go ahead and substitute the values in this question.
00:01:52.000 --> 00:01:57.080
The angle is 50 degrees, the adjacent is seven, and the hypotenuse is 𝑥.
00:01:57.280 --> 00:02:01.560
So we have the equation cos of 50 degrees equals seven over 𝑥.
00:02:03.040 --> 00:02:12.280
The first step in solving this equation for 𝑥 is to multiply both sides of the equation by 𝑥, as this will eliminate the denominator of 𝑥 on the right-hand side.
00:02:13.840 --> 00:02:17.800
We’re left with 𝑥 multiplied by cos of 50 degrees is equal to seven.
00:02:19.160 --> 00:02:26.680
Next, we need to divide both sides of this equation by cos of 50 degrees so that we’re just left with 𝑥 on its own on the left-hand side.
00:02:28.040 --> 00:02:32.280
We have then that 𝑥 is equal to seven over cos of 50 degrees.
00:02:32.280 --> 00:02:37.520
And at this point, we can use our calculator to evaluate this, making sure that it’s in degree mode.
00:02:39.000 --> 00:02:43.400
Using a calculator, we get an answer of 10.8900 continuing.
00:02:45.000 --> 00:02:47.800
Now we haven’t been asked to round our answer in a particular way.
00:02:48.160 --> 00:02:52.600
So our default degree of accuracy in this case is to round to three significant figures.
00:02:53.160 --> 00:02:57.120
The fourth significant figure is a nine, so this tells us that we’re rounding up.
00:02:58.440 --> 00:03:01.400
So the eight in the tenths column will round up to a nine.
00:03:01.760 --> 00:03:06.960
And we have that, to three significant figures, the length 𝑥 is 10.9 centimetres.
00:03:09.040 --> 00:03:12.560
Part b) of the question says, “Calculate the size of angle 𝑦.”
00:03:14.280 --> 00:03:17.640
Looking at the diagram, we can see that, again, we have a right-angled triangle.
00:03:18.120 --> 00:03:22.600
This time, we know the lengths of two sides and we want to calculate the size of one angle.
00:03:23.000 --> 00:03:25.040
So we can again apply trigonometry.
00:03:26.360 --> 00:03:28.640
As always, we begin by labelling the sides.
00:03:28.880 --> 00:03:42.120
We have the hypotenuse opposite the right angle; the opposite, which is opposite the angle we’re interested in, angle 𝑦, so that’s the side of seven centimetres; and the adjacent, which is between angle 𝑦 and the right angle.
00:03:43.640 --> 00:03:47.280
Next, we need to decide which of the three trigonometric ratios to apply.
00:03:47.760 --> 00:03:50.480
The two sides we know are the opposite and the adjacent.
00:03:50.920 --> 00:03:52.920
So we’re going to be using the tan ratio.
00:03:54.200 --> 00:03:59.840
The definition of tan is that tan of an angle 𝜃 is equal to the opposite divided by the adjacent.
00:04:00.320 --> 00:04:04.560
Let’s go ahead and substitute the values in this question.
00:04:04.560 --> 00:04:08.280
Our angle is 𝑦, the opposite is seven, and adjacent is 12.
00:04:08.640 --> 00:04:11.480
So we have tan of 𝑦 equals seven over 12.
00:04:13.400 --> 00:04:18.000
Now as we’re calculating an angle in this question, we need to apply the inverse tan function.
00:04:18.720 --> 00:04:27.040
This is the function which takes a ratio, in this case seven over 12, and tells us what angle is associated with this tan ratio.
00:04:27.520 --> 00:04:31.640
So we have that 𝑦 is equal to the inverse tan of seven over 12.
00:04:33.160 --> 00:04:35.320
We can again work this out using a calculator.
00:04:35.800 --> 00:04:39.480
The inverse tan function is located directly above the tan button.
00:04:39.960 --> 00:04:44.840
So you need to press shift and then tan in order to bring up the notation tan inverse.
00:04:45.520 --> 00:04:48.840
When we do this, we get 30.2564 continuing.
00:04:50.320 --> 00:04:53.560
Again, we haven’t been asked to give our answer to a particular degree of accuracy.
00:04:54.080 --> 00:04:55.960
So we’ll use three significant figures.
00:04:56.360 --> 00:05:01.920
The fourth significant figure is a five, which tells us we round the two in the tenths column up to a three.
00:05:02.560 --> 00:05:06.040
Remember, 𝑦 represents an angle, so the units need to be degrees.
00:05:06.920 --> 00:05:11.920
We found that the size of angle 𝑦 to three significant figures is 30.3 degrees.