Explain why the cosine law (and not the sine law) must be used when solving a triangle if the only known measures of the triangle are its three side lengths.
help
with the law of sines, you must know the angle opposite a side
the law of cosines uses all three sides, and you can solve for an angle.
thank you!
When solving a triangle using the known side lengths, the most appropriate formula to use depends on what information is available. Both the sine law and the cosine law are trigonometric formulas used to solve triangles, but they are used in different scenarios.
The sine law (also known as the law of sines) relates the lengths of the sides of a triangle to the sines of its opposite angles. It states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant:
a/sin(A) = b/sin(B) = c/sin(C)
The cosine law (also known as the law of cosines), on the other hand, relates the lengths of the sides of a triangle to the cosine of one of its angles. It states that for any triangle with sides a, b, and c, and an angle A opposite side a:
a^2 = b^2 + c^2 - 2bc*cos(A)
Now, if you have a triangle where all three side lengths are known, but none of the angles are given, you can use the cosine law to find the angles. This is because the cosine law directly relates the side lengths to the angles, allowing you to solve for the angles.
To apply the cosine law, you can choose any angle of the triangle and then use the formula to find its cosine. Once you find the value of the cosine of that angle, you can use the inverse cosine function (cos^-1) to find the measure of the angle. Repeat this process for the other angles to determine all three angles of the triangle.
In summary, when only the side lengths of a triangle are given, and you need to find the angles, you should use the cosine law since it directly relates the side lengths to the angles.