Question

question / (mandatory) (1 point)
which trigonometric tool must be used to determine the length of x in the diagram
shown?
a) the sine law
b) primary trigonometric ratios
c) the pythagorean theorem
d) two of the above are required

Explanation:

Step1: Analyze the problem

We need to find the length of \(x\). First, we use the primary trigonometric ratios (sine, cosine, tangent) in the right - angled triangle with hypotenuse \(74\) and angle \(47^{\circ}\) to find the height of the larger triangle. Then, we use the Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\)) in the smaller right - angled triangle with one side \(55\) and the height (found from the first step) to find \(x\).

Step2: Use primary trigonometric ratios

Let the height of the larger triangle be \(h\). Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), where \(\theta = 47^{\circ}\) and hypotenuse \(= 74\). So, \(h = 74\times\sin(47^{\circ})\approx74\times0.7314 = 54.1236\).

Step3: Use the Pythagorean theorem

In the smaller right - angled triangle, let \(a = 55\), \(b=x\), and \(c\approx54.1236\) (approximate value of \(h\)). By the Pythagorean theorem \(x=\sqrt{c^{2}-a^{2}}\) (Note: There is a calculation error in the problem setup assumption, actually, if we assume the height \(h\) is found correctly, and in the right - angled triangle with sides \(55\) and \(x\) and hypotenuse \(h\) (the height of the larger triangle which is perpendicular to the base of the larger triangle), we have \(x=\sqrt{h^{2}-55^{2}}\). But the key is that two tools (primary trigonometric ratios and Pythagorean theorem) are used.

Answer:

d) two of the above are required