Right Triangles and Trigonometry
This domain combines the Pythagorean theorem, special right triangles, and trigonometric ratios. The Pythagorean theorem (a^2 + b^2 = c^2, with c the hypotenuse) works only in right triangles; its converse lets you classify a triangle as right, acute, or obtuse by comparing c^2 to a^2 + b^2. Know the Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples to save time.
Special right triangles: in a 45-45-90 triangle the legs are equal and the hypotenuse is leg times the square root of 2; in a 30-60-90 triangle the sides are in ratio 1 : square root of 3 : 2, with the short leg opposite the 30-degree angle, the long leg opposite 60 degrees, and the hypotenuse twice the short leg.
Trigonometric ratios in a right triangle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. Remember SOH-CAH-TOA. Also know that sin of an angle equals cos of its complement. For angles of elevation and depression, draw the horizontal line and mark the angle from it; the angle of elevation from the ground equals the angle of depression from the top when the lines are parallel.
Traps: using the wrong leg as "opposite" when the triangle is rotated, forgetting to take the square root at the end of the Pythagorean theorem, and using the ratio backwards (multiplying instead of dividing) when solving for a side.
Worked example. A 30-60-90 triangle has hypotenuse 14. The short leg is 14 / 2 = 7, and the long leg is 7 times the square root of 3, about 12.12.