SOH CAH TOA — Right Triangle Trig

ACT Math · Axiom Academy

Right-triangle trigonometry for the ACT — the three ratios that turn a sketched triangle into a solved one. Every right triangle has three sides relative to a given acute angle : the opposite (across from ), the adjacent (next to , but not the hypotenuse), and the hypotenuse (the longest side, across from the right angle). The labels "opposite" and "adjacent" change depending on which angle you're looking at. Only the hypotenuse stays put — it's always the longest side, opposite the 90° angle. Watch the same triangle from each acute angle and see the labels swap. In a right triangle with legs 3 and 4 and hypotenuse 5, if is the angle opposite the side of length 3: 3. Example 1 — Finding a Missing Side (Sine) Problem: A right triangle has a hypotenuse of 10 and an angle of 30°. Find the side opposite the 30° angle. 4. Example 2 — Finding a Missing Side (Tangent) Problem: From the top of a 40-foot building, the angle of depression to a point on the ground is 55°. How far is the point from the base of the building? Example 3 — Finding an Angle (Inverse Trig) Problem: A right triangle has legs of length 5 and 12. Find the measure of the angle opposite the side of length 5. Label every problem with O , A , H first, then ask which two sides are involved. The pair of sides tells you the ratio — watch each pair light up and name its ratio. Problem: In a right triangle, . Find and . Example 5 — ACT-Style Multiple Choice

This is the written version of the interactive lesson above. See the full ACT Math course.