Right Triangles and Trigonometry
Pythagorean theorem and trig ratios (SOHCAHTOA) in right triangles
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Right triangles, the Pythagorean theorem, and SOHCAHTOA. The SAT asks five types of questions using these tools.
Why this matters
The test sticks to right triangles and asks five specific types of questions. Some need the Pythagorean theorem. Some need SOHCAHTOA. Some hand you a ratio and expect you to find the missing sides.
The five patterns
Direct Calculation
Use the Pythagorean theorem or SOHCAHTOA to find a missing side or trig ratio. Straightforward once you identify the right triangle.
›Ratio to Dimension
You get a trig ratio and one side length. Use them to find other sides, then calculate a perimeter or area. Two steps, one after the other.
›Special Triangles
45-45-90 and 30-60-90 triangles have fixed side-length ratios. Recognize the triangle type, apply the ratio, and the missing length drops out.
›Trig Relationships
Co-function identities: sin(x) = cos(90° − x). The SAT also tests trig properties of similar triangles. Know these shortcuts.
›Ratio Conversion
Given one trig ratio (like sin), reconstruct the triangle's sides, then find a different ratio (like tan). Build the triangle first, answer second.
The biggest trap: mixing up which side is opposite and which is adjacent. "Opposite" and "adjacent" depend on which angle you are looking at. Mislabel one side and your entire answer is wrong.