Among the cases: SSS, SAS, ASA, SSA, SAA, and AAA; Which cases can we have to solve a triangle using the Law of Sines or Cosines. Explain using the examples
The Correct Answer and Explanation is :
To solve a triangle using the Law of Sines or Cosines, it’s important to understand the given cases and their applicability. Let’s break them down:
- SSS (Side-Side-Side):
The Law of Cosines is generally used to solve triangles in the SSS case. Here, we are given the lengths of all three sides of the triangle. Using the Law of Cosines, we can find the angles. The formula is:
[
c^2 = a^2 + b^2 – 2ab \cdot \cos(C)
]
Once the angles are found, the Law of Sines can be used to determine the remaining angles and sides if needed. - SAS (Side-Angle-Side):
This case can be solved using either the Law of Cosines or the Law of Sines. First, you can use the Law of Cosines to find the third side if two sides and the included angle are known. Once the third side is determined, you can apply the Law of Sines to find the other angles. - ASA (Angle-Side-Angle):
The Law of Sines is typically used in this case. When two angles and a non-included side are known, you can use the Law of Sines to find the third angle and the remaining sides. The formula for the Law of Sines is:
[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
]
Here, you can use the angles to find the unknown sides and angles. - SSA (Side-Side-Angle):
This case is tricky and can lead to multiple solutions or no solution at all (ambiguous case). The Law of Sines can be used, but it may yield either no triangle, one triangle, or two triangles, depending on the given information. - SAA (Side-Angle-Angle):
This case is solvable using the Law of Sines, just like ASA. You can first determine the third angle (since the sum of angles in any triangle is 180°), and then use the Law of Sines to find the missing sides. - AAA (Angle-Angle-Angle):
This case doesn’t have enough information to solve the triangle uniquely. While the angles are given, the side lengths are not, so no further information can be derived from just angles.
Summary:
- Law of Cosines is used in SSS and SAS (for finding a side or angle).
- Law of Sines is used in ASA, SSA (sometimes), and SAA to find the missing angles or sides.
I’ll generate an image illustrating how to solve a triangle with the Law of Sines and Cosines.
Here is the diagram showing the Law of Sines and the Law of Cosines in action. It illustrates the relationships between the sides and angles of a triangle, which you can use to solve for unknown elements depending on the case.
