Question

1. A triangular section of a lawn will be converted to river rock instead of grass. Maurice insists that the only way to find a missing side length is to use the Law of Cosines. Johanna exclaims, that only the Law of Sines will be useful. Describe a scenario where Maurice is correct, a scenario where Johanna is correct, and a scenario where both laws are able to be used. Use complete sentences and example measurements when necessary.

Answer 1

Law of Cosines are used when an angle and two sides are given and you have to find for the other side. Law of Sines are used when you need to find the angle of the opposite side.

Answer 2

Answer:

Let's take a careful look at the triangle ABC (first attached image below). We term the sides using small letters: a, b, and c. We will also term the angles by using capital letters A, B, and C. Side a matches with angle A° it's observed to be opposing one another. We can see that those of "side b and angle B° and side c and angle C° are similar to that of side a and angle A°.

The cosine rule can be used in conditions where the length of two sides is given and the angle opposite the side is not given. Take for instance, from the second attached image, the length of side a and side b is given while the length of side c is not given as the angle positioned on the opposing of side c is given, in this case, the cosine rule applies here:

c² = a² + b² - 2ab(cos(C°))

The third attached image gives an instance where sine rule can be employed. The length of the side c wasn't given but that of the length of side a, the size of angle A° and angle C° was given. The sine rule applies here:

C/sin(C) = a/sin(A)

The sine rule can also be applied if the length of side b plus the size of angle B° is given in place of side a and angle A°.

C/sin(C) = b/sin(B)