Title 40 Solutions to Triangles.pdf ✅

MSTC 40: SOLUTIONS TO TRIANGLES 1. Sine Law It is an equation that relates the side lengths of triangles to their opposite interior angles. [DERIVATION] From trigonometry hb = c sin A hb = a sin C Transitive property of equality c sin A = a sin C c sin C = a sin A Similarly, for ha b sin C = c sin B b sin B = c sin C Transitive property of equality a sin A = b sin B = c sin C
1.1. SAA case SAA case means Side-Angle-Angle, in that consecutive order. To solve for the other parts, follow these steps Step 1: Solve for the measure of the third angle Step 2: Using sine law, solve for the other two sides 1.2. ASA case ASA means Angle-Side-Angle, in that consecutive order. To solve for the other parts, follow these steps Step 1: Solve for the measure of the third angle Step 2: Using sine law, solve for the other two sides
1.3. Ambiguous case The ambiguous case is found in the SSA case since the possible solutions may be 0 triangles, one triangle, or two possible triangles. From the given figure, here are the primary steps for solving the other parts. Checking: Solve for the height of the triangle h = x sin α. Number of possible solutions Conditions 2 Triangles x sin α < y < x 1 Triangle y ≥ x sin α 0 Triangle y < x sin α for acute α y ≤ x for obtuse α Step 1: Solve for the angle opposite to x using the sine law. There are two possible angles. If the opposite angle of x is β, then the two possible angles based on the figure are y sin α = x sin β β = { sin−1 x sin α y 180° − sin−1 x sin α y Step 2: Solve for the third angle, keeping in mind that the sum of the three interior angles of a triangle is always 180°. Also, note that you are checking for the number of solutions. Step 3: Solve for the length of the third side using the sine law.