Question

question 12 (mandatory) (1 point) which trigonometric tool must be used to determine the length of x in the diagram shown? a) the cosine law b) primary trigonometric ratios c) the pythagorean theorem d) the sine law

Explanation:

Step1: Recall trigonometric tools

• Cosine law: \(c^{2}=a^{2}+b^{2}-2ab\cos C\) (used when we know two sides and the included angle or all three sides of a non - right triangle).
• Primary trigonometric ratios (\(\sin\theta=\frac{opposite}{hypotenuse},\cos\theta=\frac{adjacent}{hypotenuse},\tan\theta=\frac{opposite}{adjacent}\)): used in right - triangles.
• Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\)): used in right - triangles to relate the sides (not involving angles).
• Sine law (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)): used in non - right triangles when we know an angle and its opposite side and another angle or side.

Step2: Analyze the given diagram

The given figure has right - triangles (the small triangles with the right - angle symbol). For a right - triangle, if we know an angle (\(39^{\circ}\) or \(49^{\circ}\)) and a side (\(2400\) km) and we want to find the length \(x\) (a side of the right - triangle), we use the primary trigonometric ratios. For example, if we consider the right - triangle with the \(39^{\circ}\) angle, and assume the side of length \(2400\) km is adjacent to the \(39^{\circ}\) angle and \(x\) is the opposite side, we can use \(\tan\theta=\frac{opposite}{adjacent}\), i.e., \(\tan39^{\circ}=\frac{x}{2400}\).

Answer:

B. primary trigonometric ratios