Question

question 5 (mandatory) (1 point) saved
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 have a right - angled triangle (the smaller one with side \(x\) and another triangle where we might need to find some lengths. First, for the non - \(x\) part:
Let's assume we first use the Pythagorean theorem in the larger right - angled triangle (the one with sides \(55\) and \(74\)). The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), but here we can find the length of the common side (let's call it \(y\)) of the two right - angled triangles. If we consider the triangle with sides \(55\) and \(74\), we can find \(y\) (the height of the larger triangle) using the formula \(y=\sqrt{74^{2}-55^{2}}\) (Pythagorean theorem: \(y^{2}+55^{2}=74^{2}\), so \(y = \sqrt{74^{2}-55^{2}}=\sqrt{(74 + 55)(74 - 55)}=\sqrt{129\times19}=\sqrt{2451}\approx49.5\)).
Then, for the smaller right - angled triangle (the one with angle \(47^{\circ}\) and side \(x\) and the side \(y\) we just (hypothetically) found using Pythagorean theorem). We use the primary trigonometric ratios. If we assume the angle \(47^{\circ}\), and if \(y\) is the adjacent side and \(x\) is the opposite side, then \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(x = y\tan47^{\circ}\) (primary trigonometric ratio: \(\tan\alpha=\frac{x}{y}\), \(x=y\tan\alpha\)).

Answer:

d) two of the above are required