Using the Law of Sines in Triangles | Step-by-Step Guide

a/sinA = b/sinB = c/sinC

The expression a/sinA = b/sinB = c/sinC is known as the Law of Sines

The expression a/sinA = b/sinB = c/sinC is known as the Law of Sines. It relates the side lengths of a triangle to the sines of its corresponding angles. In a triangle with sides of lengths a, b, and c and opposite angles A, B, and C respectively, the ratio of a side length to the sine of its opposite angle is equal to the ratio of any other side length to the sine of its opposite angle.

Let’s go through a step-by-step explanation of how to use the Law of Sines:

Step 1: Understand the Triangle
Before applying the Law of Sines, make sure you have a triangle. A triangle has three sides and three angles.

Step 2: Identify the Given Information
In the given equation, a/sinA = b/sinB = c/sinC, the letters a, b, and c represent the side lengths of the triangle, while A, B, and C represent the angles.

Step 3: Calculate the Missing Side Length or Angle
To use the Law of Sines, you need to have at least one side length and its corresponding angle. If you have two side lengths and one angle, you can calculate the missing angle. If you have two angles and one side length, you can find the missing side length.

Step 4: Set Up the Proportions
Once you have the known side lengths and angles, set up the proportions using the Law of Sines equation: a/sinA = b/sinB = c/sinC.

Step 5: Solve the Proportions
You can cross-multiply and solve the proportions to find the missing side lengths or angles.
For example, if you want to find side length b, given angle A and side lengths a and c, you can set up the proportion:
a/sinA = b/sinB
Cross-multiplying, you get: b = (sinB/a) * b = (sinA/b) * a

If you want to find angle A, given side lengths a and b and angle B, you can set up the proportion:
a/sinA = b/sinB
Cross-multiplying, you get: sinA = (a * sinB) / b
Take the inverse sine function to find angle A: A = sin^(-1)((a * sinB)/b)

Step 6: Check for Ambiguity
Remember that the Law of Sines can have two possible solutions, known as the ambiguous case, when there are multiple valid triangles that satisfy the given conditions. In such cases, you may need additional information to determine the exact triangle.

Always be careful when using the Law of Sines and ensure you have the appropriate information to solve for the missing side lengths or angles accurately.

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