Right Triangle Trig: SOH-CAH-TOA
Trigonometry connects angles to side ratios in right triangles. The mnemonic SOH-CAH-TOA helps you remember:
- SOH: sin(theta) = Opposite / Hypotenuse
- CAH: cos(theta) = Adjacent / Hypotenuse
- TOA: tan(theta) = Opposite / Adjacent
Building the Right Triangle
Use the slider to change the acute angle. Watch how the opposite and adjacent sides change while the hypotenuse stays at length 5.
The graph below shows the right triangle. The hypotenuse goes from the origin to the point on a circle of radius 5. The horizontal leg is the adjacent side and the vertical leg is the opposite side.
Try this: Set theta to 45 degrees. The opposite and adjacent sides become equal — that’s your 45-45-90 triangle! At 30 degrees, the opposite is half the hypotenuse (sin 30 = 0.5).
The Trig Ratios in Action
Now let’s see how sin, cos, and tan change as the angle grows from 0 to 90 degrees. The x-axis below represents angle theta in degrees.
Key patterns: As theta increases from 0 to 90 degrees: sin goes from 0 to 1, cos goes from 1 to 0, and tan goes from 0 toward infinity. At 45 degrees, sin = cos (the triangle is isosceles).
Special Right Triangles
Two triangles show up everywhere in math:
30-60-90 Triangle
Side ratios: 1 : sqrt(3) : 2
45-45-90 Triangle
Side ratios: 1 : 1 : sqrt(2)
Using SOH-CAH-TOA to Solve Problems
Here’s the process:
- Identify which sides you know and which you need
- Choose sin, cos, or tan based on which pair of sides is involved
- Set up the equation and solve
Challenge: A ladder leans against a wall at a 70-degree angle with the ground. The ladder is 10 feet long. How high up the wall does it reach? (Hint: the wall height is the opposite side, the ladder is the hypotenuse.)