Final answer:
To solve for the missing parts of a triangle, utilize the Law of Cosines to find an unknown angle when given two sides and an adjacent angle. Following this, the Law of Sines can be used to find additional angles or sides, keeping in mind the sum of angles in a triangle is always 180 degrees.
Step-by-step explanation:
To solve for all missing parts of a triangle when given two sides and an angle, we can use the Law of Sines and the Law of Cosines. In this specific case, if we are given a = 9, b = 12, and c = 12 where side c is adjacent to the angle we know, we would start by using the Law of Cosines to find angle C.
The Law of Cosines states: c² = a² + b² - 2ab cos y. So, if we are solving for angle C (which we shall call γ), we can rearrange the formula to get cos γ = (a² + b² - c²) / (2ab). Once we have cos γ, we can take the arccos (inverse cosine) to find the measure of angle C.
After finding angle C, we can use the Law of Sines which states:a/sin α = b/sin β = c/sin γ to find angle B. Since we now know two angles in the triangle, we can find the third angle (angle B) by subtracting the known angles from 180 degrees, because the sum of angles in any triangle is 180 degrees. For side b, we already have this value as one of the provided variables. Hence, we have all the missing parts of the triangle.