Question

determining when to use the law of cosines
which of these triangles can you use the law of cosines to solve for a missing side?
(images of four triangles: first with angles 75°, 38° and side 8, missing side x; second with angle 138°, sides 12 and 8, missing side x; third right triangle with angle 35°, missing sides x and y; fourth equilateral-like with angles 60°, 60°, 60°, missing side x)

Explanation:

Step1: Recall the Law of Cosines formula

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\), where \(a\), \(b\), \(c\) are the sides of the triangle and \(C\) is the angle opposite to side \(c\). It is used when we know two sides and the included angle or all three sides of a triangle.

Step2: Analyze the first triangle

In the first triangle, we know two angles (\(75^{\circ}\) and \(38^{\circ}\)) and one side. We can first find the third angle (\(180-(75 + 38)=67^{\circ}\)) and then use the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)) since we have angle - side - angle information.

Step3: Analyze the second triangle

In the second triangle, we know two sides (\(12\) and \(8\)) and the included angle (\(138^{\circ}\)). This is the \(SAS\) (side - angle - side) case. For the \(SAS\) case, we can use the Law of Cosines. Let \(a = 12\), \(b=8\), \(C = 138^{\circ}\), and we want to find \(x\). Then \(x^{2}=12^{2}+8^{2}-2\times12\times8\times\cos(138^{\circ})\).

Step4: Analyze the third triangle

In the third triangle, it is a right - triangle. We can use trigonometric ratios (\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)) or the Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\)) instead of the Law of Cosines.

Step5: Analyze the fourth triangle

In the fourth triangle, since all angles are \(60^{\circ}\), it is an equilateral triangle (\(x\) is equal to the other sides). We don't need the Law of Cosines. We know that in an equilateral triangle all sides are equal.

Answer:

The second triangle.