Question

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 triangle structure

We have two right - angled triangles. First, we can use the primary trigonometric ratio (sine ratio) in the triangle with side \(74\) and angle \(47^{\circ}\) to find the common side (let's call it \(h\)) between the two right - angled triangles. The sine ratio is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(h = 74\sin47^{\circ}\).

Step2: Use the Pythagorean theorem

In the other right - angled triangle with sides \(x\), \(55\) and the side \(h\) (calculated above), we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse). Here, \(x=\sqrt{h^{2}-55^{2}}\) (after substituting \(h = 74\sin47^{\circ}\) into the formula).

Answer:

d) two of the above are required