ACT® · Math: Geometry — Similarity, Right Triangles, and Trigonometry
Sine, Cosine and Tangent as Side Ratios
In a right triangle, sine, cosine and tangent of an acute angle are fixed ratios between two of its sides. The angle alone determines these ratios, no matter the triangle's actual size.
Pick an acute angle in a right triangle and stand at its vertex, looking across the triangle. The side across from you is the opposite side. The side touching your angle that isn't the hypotenuse is the adjacent side. The hypotenuse is always the longest side, opposite the right angle, and it never changes identity no matter which acute angle you choose. Switch to the other acute angle and opposite and adjacent swap places, since the side that was adjacent is now the one across from you.
SOHCAHTOA packages this into three fractions. Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. Once you fix an angle, these ratios are locked in by the angle's size alone. A 3-4-5 triangle and a 30-40-50 triangle have the same angles, so they give the same sine, cosine and tangent values, even though their side lengths differ. That's the whole reason trig ratios are useful: the angle alone tells you the shape of the triangle, and any similar triangle sitting on that angle gives the same three fractions.
This sits right next to the two special right triangles, 45-45-90 and 30-60-90, where you get to memorize the ratios instead of computing them from a Pythagorean triple. It also sets up the laws of sines and cosines, which extend these same ideas to triangles without a right angle, once you no longer have opposite, adjacent and hypotenuse to lean on.
Key terms
- Opposite side
- The side across from the acute angle you're focused on, not touching that angle.
- Adjacent side
- The leg that touches the acute angle you're focused on but is not the hypotenuse.
- Hypotenuse
- The longest side of a right triangle, opposite the right angle, the same for either acute angle.
- SOHCAHTOA
- A memory device: Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent.
Watch it explained
- How it worksHow do opposite, adjacent and hypotenuse sort themselves out once you choose which acute angle to stand at?
- How it worksHow does SOHCAHTOA turn a named angle into three specific fractions of side lengths?
- Worked exampleFind the sine, cosine and tangent of the smaller acute angle in a right triangle with legs 8 and 15.
- Going furtherWhat happens to all three ratios when you switch your attention to the other acute angle?
Check yourself
End of topic.