PSAT/NMSQT Math · Lesson 4 of 4
Geometry and right-triangle trigonometry
Use exact relationships among lengths and angles, and attach each trigonometric ratio to a chosen angle.
Diagrams
Write given lengths on the figure and mark right angles as given, not as guessed. Parallel lines create equal corresponding angles. Those equal angles can show that triangles are similar, and then corresponding sides are proportional.
Area and volume formulas need the dimension they name. The area of a triangle uses a base and the height perpendicular to that base. The volume of a prism is the area of the base times the perpendicular height of the prism.
Trigonometric ratios
Pick the acute angle you are using and label the opposite side, the adjacent side, and the hypotenuse. Sine is opposite over hypotenuse. The same triangle gives a different sine for the other acute angle.
If the hypotenuse and one acute angle are known, a sine or cosine equation produces a side. If two sides of a right triangle are known, the Pythagorean theorem produces the third side without trigonometry.
Worked example
A ramp rises 3 units over a horizontal distance of 4 units. How long is the ramp, and what is the sine of the angle it makes with the ground?
- The ramp is the hypotenuse: √(3² + 4²) = 5.
- The angle with the ground has opposite side 3 and hypotenuse 5.
- Sine of that angle is 3/5.
Why this works. The hypotenuse is the ramp itself, not the horizontal distance.
Check your understanding
- Mark right angles and parallel lines only when the problem supports them.
- Use perpendicular height in area and volume formulas.
- Re-label opposite and adjacent when the reference angle changes.