Trigonometry from zero
No trig background needed. By the end of this page you'll know what sin, cos and tan are, why they work, and how to use them to find a side or an angle you can't measure directly.
1. What is trigonometry for?
Say you want the height of a tree. You can't climb it with a tape measure, but you can measure its shadow on the ground and the angle the sun makes. Trigonometry turns those two easy measurements into the hard one.
It works because of one fact about right triangles (triangles with a 90° corner): once you know one of the other angles, the shape is fixed. Only the size can change. So the angle tells you how the sides compare to each other, and one known side tells you the size.
Everything here uses right triangles only. That's where trig starts, and it covers ladders, ramps, roofs, shadows, slopes and distances.
2. Name the sides
A right triangle has three sides, and trig gives each one a name. One name is permanent; the other two depend on which angle you're looking from.
- Hypotenuse: the longest side, across from the right angle. It never changes.
- Opposite: the side across from the angle you're looking from. It doesn't touch that angle.
- Adjacent: the other side that touches your angle ("adjacent" means "next to"). It's not the hypotenuse.
3. The big idea: same angle, same ratios
Move the sliders. Change the size and every side gets longer or shorter, but the side lengths divided by each other (the ratios) stay the same. Change the angle and the ratios change.
Each of those three ratios depends only on the angle, so mathematicians gave them names:
"SOH CAH TOA" (say it "so-ka-toe-uh") is the standard way to remember which is which. Your calculator knows these ratios for every angle: type sin(35) and it tells you that for a 35° angle, the opposite side is about 0.5736 times the hypotenuse.
θ (the Greek letter "theta") is just the usual name for an angle, like x for a number.
4. The four-step recipe
Every right-triangle problem is solved the same way:
- Label the sides from the angle you're given (or the angle you want): opposite, adjacent, hypotenuse.
- Pick the ratio that uses the two sides in play: the one you know and the one you want. SOH CAH TOA tells you which.
- Write the equation, like sin(35°) = x / 12.
- Solve it with ordinary algebra and a calculator.
Here's the recipe on a real problem. Each example is freshly generated, so press Another example as often as you like.
5. When x lands on the bottom
Sometimes the side you want is the one the ratio divides by, like cos(40°) = 9 / x. Don't panic: multiply both sides by x to get it out of the bottom, then divide by the trig value:
Quick sense check: the hypotenuse is always the longest side, so if you're finding a hypotenuse your answer must be bigger than the other sides.
6. Finding an angle: run it backwards
If you know two sides and want the angle, you know the ratio and want the angle that makes it. That's what the inverse buttons do: sin⁻¹, cos⁻¹ and tan⁻¹ (often labeled 2nd + sin, or asin).
Read tan⁻¹(0.7) as "the angle whose tan is 0.7". The ⁻¹ means "undo", not "divide by".
Calculator check: degree mode
Calculators measure angles in either degrees or radians (a different unit you'll meet later). All the problems here use degrees. Test yours: sin(30) should give 0.5. If you get −0.988, it's in radian mode: look for a DEG/RAD setting. The practice app spots radian-mode answers and tells you.
Ready to practice?
Five levels, building one skill at a time. Every problem shows its full solution step by step, with the triangle labeled, after you answer.
- Write sin, cos or tan as a fraction (just labeling and SOH CAH TOA)
- Find a side when x is on top of the ratio
- Find a side when x is on the bottom
- Find an angle with inverse trig
- Word problems: ladders, ramps, kites and shadows
Next: the unit circle and radians
SOH CAH TOA only works for angles under 90°. The unit circle extends sin and cos to every angle, and introduces radians, the other way to measure angles.