What does SOHCAHTOA mean?

SOHCAHTOA is a memory aid for the three right-triangle ratios. SOH: sine equals opposite over hypotenuse. CAH: cosine equals adjacent over hypotenuse. TOA: tangent equals opposite over adjacent. These three ratios let you find any missing side or angle in a right triangle.

How do you use SOHCAHTOA to find a missing side?

Choose the ratio that links your known angle, your known side and the side you want. If you know the angle and hypotenuse and want the opposite side: opposite = hypotenuse x sin(angle). For angle 30 and hypotenuse 10: opposite = 10 x sin(30) = 10 x 0.5 = 5.

Which sides are opposite, adjacent and hypotenuse?

The hypotenuse is the longest side, always opposite the right angle. For your chosen angle, the opposite side is across from it and the adjacent side is next to it but not the hypotenuse.

SOHCAHTOA Solver

SOHCAHTOA is the famous memory aid for right-angle trigonometry: SOH means sine equals opposite over hypotenuse, CAH means cosine equals adjacent over hypotenuse, and TOA means tangent equals opposite over adjacent. Between them, these three ratios connect any angle in a right triangle to any two of its sides. If you know one angle and one side, every other measurement in the triangle is determined. This calculator puts that to work: enter the angle, tell it which side you know (the opposite, the adjacent, or the hypotenuse), enter its length, and it solves all three sides and the other acute angle instantly. The right angle is always the third angle, so the two acute angles add to 90 degrees, which means knowing one gives you the other automatically. The calculator picks the right ratio for each case. If you know the hypotenuse: opposite = hypotenuse times sine, adjacent = hypotenuse times cosine. If you know the opposite: hypotenuse = opposite divided by sine, adjacent = opposite divided by tangent. If you know the adjacent: hypotenuse = adjacent divided by cosine, opposite = adjacent times tangent. Seeing these results calculated for your specific inputs makes the logic clear and concrete, which is why this tool is well suited to students working through trigonometry for the first time. It is also genuinely useful for practical problems in construction, surveying, and navigation, where you know a slope angle and one measurement and need the rest. Enter any angle between 0 and 90 degrees exclusive and any positive side length, and the results update instantly.

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Angle (degrees)

Known side

Side length

units

opp 5 · adj 8.66 · hyp 10

solved triangle sides

Opposite5

Adjacent8.66

Hypotenuse10

Other angle60°

How it works

The three ratios: SOH (sin = opp/hyp), CAH (cos = adj/hyp), TOA (tan = opp/adj). From angle and hypotenuse: opp = hyp × sin, adj = hyp × cos. From angle and opposite: hyp = opp / sin, adj = opp / tan. From angle and adjacent: hyp = adj / cos, opp = adj × tan. The other acute angle is always 90 degrees minus the given angle.

Worked example

With angle = 30 degrees and hypotenuse = 10: opposite = 10 × sin(30) = 10 × 0.5 = 5. Adjacent = 10 × cos(30) = 10 × 0.866 = 8.66. Hypotenuse = 10. Other angle = 90 minus 30 = 60 degrees. These match the pre-filled defaults above.

Related calculators

Unit Circle Calculator: sine, cosine and tangent for any angle on the unit circle with a 16-angle reference table.
Inverse Trig Calculator: find arcsin, arccos and arctan from a ratio value.
Reference Angle Calculator: find the reference angle for any angle in any quadrant.
Law of Sines and Cosines: solve non-right triangles from sides and angles.

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