Part a) Calculate the length 𝑥.
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. 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.
However, if we wanted to use Pythagoras theorem, we need to know both of the other two side lengths. And from the diagram, we can see that we’re only given one side length. We’re also given one of the other angles apart from the right angle. So this tells us that we’re going to need to use trigonometry for this question.
The first step in any problem involving trigonometry is to label the three sides of the triangle. First, we have the hypotenuse, H, the longest side of the triangle, which is always directly opposite the right angle. We then have the opposite side, O, which is opposite the other angle that we’ve been given. In this case, that’s the angle of 50 degrees. And finally, we have the adjacent side, A, which is between the right angle and the angle we’ve been given.
We can then use the memory aid SOHCAHTOA to help us decide which of the three trigonometric ratios we need in this question. Here, S, C, and T stand for sin, cos, and tan and O, A, and H stand for opposite, adjacent, and hypotenuse.
The side we know is the adjacent, and the side we want to calculate is the hypotenuse. So the pair of sides that are involved in the ratio are A and H, which is the cos ratio. Let’s recall its definition. Cos or cosine of an angle 𝜃 is equal to the length of the adjacent side divided by the length of the hypotenuse.
We can go ahead and substitute the values in this question. The angle is 50 degrees, the adjacent is seven, and the hypotenuse is 𝑥. So we have the equation cos of 50 degrees equals seven over 𝑥. 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. We’re left with 𝑥 multiplied by cos of 50 degrees is equal to seven.
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. We have then that 𝑥 is equal to seven over cos of 50 degrees. And at this point, we can use our calculator to evaluate this, making sure that it’s in degree mode. Using a calculator, we get an answer of 10.8900 continuing.
Now we haven’t been asked to round our answer in a particular way. So our default degree of accuracy in this case is to round to three significant figures. The fourth significant figure is a nine, so this tells us that we’re rounding up. So the eight in the tenths column will round up to a nine. And we have that, to three significant figures, the length 𝑥 is 10.9 centimetres.
Part b) of the question says, “Calculate the size of angle 𝑦.”
Looking at the diagram, we can see that, again, we have a right-angled triangle. This time, we know the lengths of two sides and we want to calculate the size of one angle. So we can again apply trigonometry.
As always, we begin by labelling the sides. 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.
Next, we need to decide which of the three trigonometric ratios to apply. The two sides we know are the opposite and the adjacent. So we’re going to be using the tan ratio. The definition of tan is that tan of an angle 𝜃 is equal to the opposite divided by the adjacent.
Let’s go ahead and substitute the values in this question. Our angle is 𝑦, the opposite is seven, and adjacent is 12. So we have tan of 𝑦 equals seven over 12. Now as we’re calculating an angle in this question, we need to apply the inverse tan function. 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. So we have that 𝑦 is equal to the inverse tan of seven over 12.
We can again work this out using a calculator. The inverse tan function is located directly above the tan button. So you need to press shift and then tan in order to bring up the notation tan inverse. When we do this, we get 30.2564 continuing.
Again, we haven’t been asked to give our answer to a particular degree of accuracy. So we’ll use three significant figures. The fourth significant figure is a five, which tells us we round the two in the tenths column up to a three. Remember, 𝑦 represents an angle, so the units need to be degrees. We found that the size of angle 𝑦 to three significant figures is 30.3 degrees.