Pre-Calculus Prep: Law of Sines - Solve the Triangle

Pre-Calculus Prep: Law of Sines - Solve the Triangle

In pre-calculus, the Law of Sines is a powerful tool used to solve triangles that are not right triangles. It relates the ratios of the lengths of the sides to the sines of their opposite angles. The Law of Sines states:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the measures of the angles opposite those respective sides.

To solve a triangle using the Law of Sines, you need to know either two angles and one side, or two sides and one angle. With this information, you can use the Law of Sines to find the remaining side lengths and angle measures. Here's a step-by-step approach to solving a triangle using the Law of Sines:

1. Identify the given information: Look for the known side lengths and angle measures in the triangle. You need to have at least two sides and one angle, or two angles and one side, to apply the Law of Sines.

2. Determine which ratios to use: Identify the ratios of side lengths to sines of angles that are available based on the given information. Choose the appropriate ratios based on what you need to find.

3. Set up the Law of Sines equation: Write down the Law of Sines equation using the chosen ratios. For example, if you want to find the length of side a, you would write:

a/sin(A) = b/sin(B) = c/sin(C)

4. Solve for the unknowns: Use algebraic manipulation to solve the equation for the unknown side length(s) or angle measure(s). If you're solving for a side length, cross-multiply the ratios and isolate the desired side length. If you're solving for an angle measure, use inverse trigonometric functions to isolate the desired angle.

5. Calculate the missing values: Substitute the known values into the equation and use a calculator or trigonometric tables to evaluate the expression. Round your answers to the desired level of precision.

6. Verify the triangle: Check if the calculated side lengths and angle measures satisfy the triangle inequality and the sum of angles in a triangle (which is 180 degrees). If they don't, double-check your calculations and make corrections if necessary.

By following these steps and applying the Law of Sines, you can successfully solve a triangle, finding the missing side lengths and angle measures in pre-calculus.
These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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