Today we will explore the special relationship between the three sides of an isosceles triangle with a fifteen-degree angle. In triangle ABC, sides AB and AC are equal, and angle A measures fifteen degrees. We want to find the relationship between the equal sides a and the base b.
To find the relationship between the sides, we use the Law of Cosines. For triangle ABC with angle A equal to fifteen degrees and equal sides a, the formula becomes: b squared equals a squared plus a squared minus two a times a times cosine of fifteen degrees. This simplifies to b squared equals two a squared times one minus cosine of fifteen degrees.
Now we need to calculate cosine of fifteen degrees. We can use the angle subtraction formula: cosine of fifteen degrees equals cosine of forty-five degrees minus thirty degrees. Using the cosine difference formula, this becomes cosine of forty-five degrees times cosine of thirty degrees plus sine of forty-five degrees times sine of thirty degrees. Substituting the known values, we get square root of two over two times square root of three over two plus square root of two over two times one half, which equals square root of six plus square root of two, all over four.
Now we substitute the value of cosine fifteen degrees back into our equation. We have b squared equals two a squared times one minus square root of six plus square root of two over four. This simplifies to b squared equals two a squared times four minus square root of six minus square root of two, all over four. Further simplifying, we get b squared equals a squared times four minus square root of six minus square root of two, all over two. Taking the square root of both sides, we obtain the final relationship: b equals a times the square root of four minus square root of six minus square root of two, all over two.
To summarize what we have learned: In a fifteen-degree isosceles triangle with equal sides a, the base b is related to the equal sides through a specific formula. The base equals a times the square root of four minus square root of six minus square root of two, all divided by two. This relationship was derived using the Law of Cosines and gives us a ratio of approximately zero point five one eight between the base and the equal sides.