Trigonometry Made Easy: Solve Any Triangle with SOH CAH TOA and Beyond
Solve any trigonometry problem with our comprehensive guide. Learn SOH CAH TOA, law of sines, law of cosines, and how to find missing sides and angles.
Trigonometry is the mathematics of triangles and angles. Whether you are solving a simple right triangle using SOH CAH TOA or tackling a complex oblique triangle with the law of sines and cosines, the principles of trigonometry allow you to find any missing measurement given enough information. This guide covers everything from basic ratios to advanced triangle-solving techniques.
Key Takeaway
For right triangles, use SOH CAH TOA ratios (sin = opp/hyp, cos = adj/hyp, tan = opp/adj). For general triangles, use the law of sines (for AAS, ASA, SSA) or law of cosines (for SSS, SAS). Always check that angles sum to 180°.
1. Right Triangle Trigonometry (SOH CAH TOA)
In a right triangle, one angle is always 90°. The side opposite the right angle is the hypotenuse (the longest side). The other two sides are the opposite and adjacent sides relative to one of the acute angles.
Solving a Right Triangle
Given one side and one acute angle, you can find the remaining two sides and the other angle. For example, if angle A = 35° and hypotenuse = 10: opposite = 10·sin(35°) ≈ 5.74, adjacent = 10·cos(35°) ≈ 8.19, and angle B = 90° - 35° = 55°.
2. Law of Sines (AAS, ASA, SSA)
Use the law of sines when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA). The SSA case is the ambiguous case, which can produce 0, 1, or 2 solutions.
3. Law of Cosines (SSS, SAS)
Use the law of cosines when you know three sides (SSS) and need to find angles, or two sides and the included angle (SAS) and need the third side. This formula generalizes the Pythagorean theorem to non-right triangles.
4. Special Right Triangles
| Triangle | Side Ratio | Trig Values |
|---|---|---|
| 30-60-90 | 1 : √3 : 2 | sin30=0.5, cos30≈0.866 |
| 45-45-90 | 1 : 1 : √2 | sin45≈0.707, tan45=1 |
5. Frequently Asked Questions
When do I use law of sines vs law of cosines?
Law of sines for AAS, ASA, SSA. Law of cosines for SSS, SAS. If you know two angles, use law of sines. If you know three sides, use law of cosines.
What is the ambiguous case?
SSA (two sides and a non-included angle) can produce 0, 1, or 2 valid triangles. Check the relationship between the given angle, opposite side, and adjacent side to determine the number of solutions.
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