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 problem

We need to find the length of \(x\). First, we use the primary trigonometric ratios (sine, cosine, tangent) in the right - triangle with side \(34\) and angle \(47^{\circ}\) to find the length of the common side (let's call it \(h\)) between the two right - triangles. Then we use the Pythagorean theorem in the right - triangle with sides \(5\) and \(h\) to find \(x\).

Step2: Use the primary trigonometric ratios

In the right - triangle with hypotenuse \(34\) and angle \(47^{\circ}\), if we consider the sine ratio \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Let the opposite side (the common side) be \(h\). So \(h = 34\sin47^{\circ}\approx34\times0.7314 = 24.8676\)

Step3: Use the Pythagorean theorem

In the right - triangle with sides \(5\) and \(h\) (where \(h\approx24.8676\)) and hypotenuse \(x\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), we have \(x=\sqrt{h^{2}-5^{2}}\) (since \(h\) is the hypotenuse of the smaller right - triangle with one side \(5\)). Substitute \(h\approx24.8676\) into the formula: \(x=\sqrt{(24.8676)^{2}-5^{2}}=\sqrt{618.39 - 25}=\sqrt{593.39}\approx24.36\)

Answer:

d) two of the above are required