Oblique Triangles: Step-by-Step Guide
An oblique triangle is any triangle that does not have a right angle. To solve one, you find all three sides and all three angles. You use the Law of Sines or the Law of Cosines, depending on what you know to start with.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Figure out what you know. Is it AAS, ASA, SSA, SAS, or SSS?\[\text{AAS, ASA, SSA, SAS, SSS}\]
- Pick the right law. Use the Law of Sines for AAS, ASA, or SSA. Use the Law of Cosines first for SAS or SSS.\[\frac{a}{\sin A} = \frac{b}{\sin B},\;\; c^2 = a^2 + b^2 - 2ab\cos C\]
- Find the first unknown. Use your law to find the easiest missing side or angle.\[x = \frac{a \sin X}{\sin A}\]
- Use the angle sum and keep going. The angles add up to 180 degrees. Use that, along with the other law, to find the rest.\[A + B + C = 180^\circ\]
- Watch for the ambiguous case (SSA). With SSA, check whether 0, 1, or 2 triangles fit.\[B' = 180^\circ - B\]
- Check the whole triangle. Make sure the angles add up to 180 degrees and the sides follow the triangle inequality.\[a + b > c,\;b + c > a,\;a + c > b\]
Worked example
Solve the triangle: a = 10, b = 14, C = 70 deg.
Step 1: This is SAS (two sides and the angle between them).
Step 2: Use the Law of Cosines first.
Step 3: c^2 = 100 + 196 - 2 x 10 x 14 x cos(70 degrees) = 296 - 280 x 0.342 = 296 - 95.76 = 200.24. So c is about 14.15.
Step 4: Use the Law of Sines: sin(A) / 10 = sin(70 degrees) / 14.15. sin(A) = 10 x 0.9397 / 14.15 = about 0.6641, so A is about 41.6 degrees. Then B = 180 - 70 - 41.6 = 68.4 degrees.
Step 5: This was not SSA, so there is no ambiguity.
Step 6: Check: 41.6 + 68.4 + 70 = 180 degrees. The triangle inequality holds. It works.
The answer is c about 14.15, A about 41.6 degrees, and B about 68.4 degrees.
Tips and common mistakes
- AAS or ASA: use the Law of Sines. There is always exactly one triangle.
- SSA: use the Law of Sines and check for the ambiguous case.
- SAS: use the Law of Cosines for the missing side first, then the Law of Sines.
- SSS: use the Law of Cosines for the biggest angle first, then the Law of Sines.
- Draw the triangle and label each side next to the angle across from it.
- When you use the Law of Sines to find an angle, find the smallest unknown angle first. This avoids getting a wrong obtuse angle.
- Check with the angle sum (180 degrees) and the triangle inequality at the end.
- Keep a few extra decimals in the middle of the problem to avoid rounding errors.