A roofer calling out a 7/12 pitch is speaking trigonometry without ever saying the word. For every 12 inches of horizontal run, the roof rises 7 inches, and the roof's angle is arctan(7/12), about 30.26 degrees. A ladder crew, a surveyor, a game developer, and a wheelchair-ramp builder are all solving the same puzzle in different clothes: a right triangle with one angle and one side known, and everything else to find. The three workhorse ratios that solve it are sine, cosine, and tangent, packed into the mnemonic every US geometry student learns: SOH-CAH-TOA. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

The first skill is naming the sides, and the names depend on where you stand. Pick one acute angle as your reference angle, theta. The hypotenuse is always the longest side, across from the right angle, and it never changes its name. The opposite side is the one across from your reference angle, not touching it. The adjacent side is the one next to your reference angle that is not the hypotenuse. Switch your reference angle to the other acute corner and the opposite and adjacent sides trade places, which is the single most common source of wrong answers in trigonometry homework. Draw the triangle, mark the reference angle with a dot or a small arc, and label the three sides before you choose a ratio. Thirty seconds of labeling saves ten minutes of untangling.

Choosing the ratio is then mechanical: look at which two sides you know or want, and pick the ratio that uses exactly those two. Opposite and hypotenuse means sine. Adjacent and hypotenuse means cosine. Opposite and adjacent means tangent. If you know an angle and want a side, you multiply: side equals the other side times the ratio. If you know two sides and want the angle, you divide and then use the inverse function: arcsin, arccos, or arctan, the shift or second-function buttons on a calculator. This calculator handles the multiplication step at the heart of each ratio: enter a known side and the reciprocal of the other side, and the result is the ratio itself, ready for the inverse button or the next step of the problem.

Work the ladder problem all the way through, because it contains every move. A 20-foot ladder leans against a wall with its base 6 feet from the wall. How high does it reach, and what angle does it make with the ground? The ladder is the hypotenuse, 20 feet. The ground distance is the adjacent side, 6 feet, relative to the angle at the ground. The height is the opposite side, unknown. Pythagoras finds it first: height equals the square root of 400 minus 36, the square root of 364, which is 19.08 feet. The angle comes from cosine: cos(theta) equals adjacent over hypotenuse, 6 over 20, which is 0.3, and arccos(0.3) is 72.54 degrees. Check it with sine: opposite over hypotenuse is 19.08 over 20, which is 0.954, and arcsin(0.954) is also 72.54 degrees. Two routes, one answer, which is how you know the work is right. Note the safety lesson hiding in the numbers: at 72.54 degrees the ladder is steep, near the top of the safe range, and a base much closer to the wall would tip it backward.

Now measure something you cannot climb: a tree. Stand 50 feet from the trunk, sight the top, and read an angle of elevation of 38 degrees on a clinometer app. The distance is the adjacent side, the tree's height above your eye is the opposite side, and tangent connects them: opposite equals adjacent times tan(theta), so 50 times tan(38 degrees), and tan(38 degrees) is 0.7813, giving 39.06 feet. Add your eye height, 5.5 feet, and the tree stands 44.56 feet tall. Forgetting the eye height is the classic error in this problem, and it is a 5.5-foot error, not a rounding error. Surveyors do exactly this with fancier instruments, and foresters estimate timber volume from the same height. The same setup measures flagpoles, buildings, and cliffs: one distance, one angle, one tangent, one addition.

Ramps and roofs are the same triangle wearing work boots. A wheelchair ramp that rises 2.5 feet over a horizontal run of 30 feet makes an angle of arctan(2.5 over 30), which is arctan(0.0833), or 4.76 degrees. That is no coincidence: the ADA guideline of a 1:12 slope ratio, one foot of rise per twelve feet of run, is exactly arctan(1 over 12), which is 4.76 degrees. The guideline is a trigonometry answer written as a building code. Roofers work the other direction: given the pitch as a rise-over-run fraction, the angle is arctan of that fraction, and the rafter length is the hypotenuse. A 7/12 roof has an angle of arctan(7 over 12), 30.26 degrees, and over a 12-foot horizontal run the rafter length is the square root of 144 plus 49, the square root of 193, which is 13.89 feet before overhang. A roofer who knows the pitch can cut the rafter without ever climbing up to measure it.

The skate-ramp version shows the angle-to-side direction. A ramp surface 14 feet long set at 20 degrees to the ground rises to a height of 14 times sin(20 degrees). Sin(20 degrees) is 0.3420, so the deck height is 4.79 feet, and the horizontal run is 14 times cos(20 degrees), cos(20 degrees) being 0.9397, giving 13.16 feet of floor space. Builders check the floor space first, because a ramp that fits the height but not the room is a ramp that does not get built. Video game physics runs the same two multiplications thousands of times a second: given a speed and a direction angle, the horizontal and vertical components are the speed times cos and sin of the angle, which is how characters move diagonally at the right speed instead of too fast.

Three special angles are worth memorizing, because they appear constantly and their exact values double as a calculator check. At 30 degrees, sin is 0.5, cos is 0.8660, tan is 0.5774. At 45 degrees, sin and cos are both 0.7071, and tan is exactly 1, which is why a 45-degree slope rises exactly as far as it runs. At 60 degrees, sin is 0.8660, cos is 0.5, tan is 1.7321. If your calculator gives sin(30 degrees) as anything but 0.5, you are in radian mode, and every answer that follows will be wrong. Degrees versus radians is the silent killer of trigonometry homework: school problems use degrees, calculus and programming often use radians, and 180 degrees equals pi radians, so 45 degrees is 0.7854 radians. Check the mode indicator, DEG versus RAD, before the first calculation, not after the tenth.

Inverse functions have one guardrail worth knowing: sine and cosine ratios can never exceed 1, because the hypotenuse is the longest side. If your division gives an opposite-over-hypotenuse ratio of 1.2, you have mislabeled the sides, swapped the hypotenuse with a leg, or divided upside down. No angle has a sine of 1.2, and your calculator will answer with an error rather than a number. Tangent has no such ceiling, since the opposite side can be longer than the adjacent side, which is why steep roofs have tangents above 1 and gentle ramps have tangents near zero. The reciprocal ratios, cosecant, secant, and cotangent, are simply one over sine, cosine, and tangent, and they show up in physics and engineering more than in school, but they are the same three triangles viewed upside down.

Angles of depression are the mirror image of angles of elevation, and they work the same way. Stand on an 80-foot cliff and sight a boat with a clinometer reading a depression angle of 25 degrees below horizontal. The height is the opposite side, the horizontal distance is the adjacent side, and distance equals height divided by tan(theta): 80 divided by tan(25 degrees), tan(25 degrees) being 0.4663, gives 171.56 feet to the boat. The neat fact that makes this problem fair is that the depression angle from the cliff equals the elevation angle from the boat, alternate interior angles of the parallel horizontal lines, so the same triangle solves it from either end. Coast guards, drone pilots judging ground distance from altitude, and skiers estimating the runout below a lift tower all use this version of the triangle.

To use this calculator, label your triangle first, decide which ratio fits the sides you know, and enter the known side in the Variable A box and one divided by the other side in the Variable B box. For the roof example, entering 7 and one divided by 12 gives 0.5833, the tangent of the pitch, and arctan of that is 30.26 degrees. For the ladder, entering 6 and one divided by 20 gives 0.3, the cosine of the ground angle, and arccos of that is 72.54 degrees. Then verify with Pythagoras or with a second ratio, as the ladder example does, because two routes to the same angle are worth more than one. If you need the sine, cosine, or tangent of an angle directly, the special-angle values above are your check figures, and the unit circle, the circle of radius 1 where every angle lands on the point (cos theta, sin theta), is the same idea extended to every angle. For triangles that are not right triangles, the law of sines and the law of cosines take over where SOH-CAH-TOA stops, and the triangle calculators on this site cover those cases.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Trigonometry Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.