SOH CAH TOA is the rule for choosing which trig ratio to use in a right-angled triangle. Label the two shorter sides opposite and adjacent relative to the marked angle, and the longest side hypotenuse. Then pick the ratio that contains the side you know and the side or angle you want: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. To find a side, multiply or divide directly. To find an angle, use the inverse function (shift sin, cos, or tan on your calculator).
SOH CAH TOA looks like three unrelated formulas to memorise, so students often guess which one to use instead of working it out. The real problem is usually earlier than the ratio: the side labels (opposite, adjacent, hypotenuse) are relative to whichever angle is marked in the diagram, not fixed to a position in the triangle. Rotate the triangle or mark a different angle, and the same physical side can flip from opposite to adjacent.
Get the labelling right first and the ratio choice becomes mechanical: look at the two sides involved (the one you know and the one you want), and pick whichever of SOH, CAH, TOA contains exactly that pair.
Step 1
Mark the given angle. Label hypotenuse (opposite the right angle), then opposite (across from the marked angle) and adjacent (next to it, not the hypotenuse)
Step 2
Identify the two sides you know or want: opp+hyp, adj+hyp, or opp+adj
Step 3
Match the pair to SOH, CAH, or TOA
Step 4
Side missing: multiply or divide. Angle missing: use shift sin / shift cos / shift tan
In right-angled triangle ABC, the right angle is at B. Angle A = 42° and the side adjacent to A, AB = 9.4 cm. Find the length of BC, the side opposite A, to 3 significant figures.
Solution
BC is opposite, AB is adjacent, AC is the hypotenuse.tan A = opposite / adjacent, so tan 42° = BC / 9.4.BC = 9.4 × tan 42° = 9.4 × 0.9004 = 8.4638...BC = 8.46 cm (3 s.f.)Quick check: the opposite side should be shorter than the adjacent side when the angle is under 45°, and longer when the angle is over 45°. Here 42° is just under 45°, so BC = 8.46 cm being slightly shorter than AB = 9.4 cm makes sense.
In right-angled triangle PQR, the right angle is at Q. The side opposite P is QR = 7.2 cm and the side adjacent to P is PQ = 5.5 cm. Find angle P to 1 decimal place.
Solution
QR is opposite, PQ is adjacent.tan P = opposite / adjacent = 7.2 / 5.5.tan P = 1.3091 (4 d.p.)P = tan⁻¹(1.3091). On your calculator this is shift then the tan key.P = 52.6° (1 d.p.)The step students get wrong
Two mistakes cause most lost marks here. First, mislabelling opposite and adjacent: both are "next to" the right angle in some sense, so always check which one is directly across from the marked angle (that one is opposite) and which one forms the angle together with the hypotenuse (that one is adjacent). Second, calculator mode: if your calculator is set to radians instead of degrees, tan 42° and tan⁻¹(1.3091) both come out completely wrong, often without looking obviously wrong. Check the mode indicator (usually a small "D" for degrees) at the top of the screen before every trig question.
How do I know which side is opposite and which is adjacent?
Both depend entirely on which angle is marked. The hypotenuse never changes: it is always the longest side, opposite the right angle. Of the remaining two sides, the one directly across from the marked angle is opposite, and the one touching the marked angle (that is not the hypotenuse) is adjacent. Re-mark a different angle in the same triangle and opposite and adjacent swap.
What if I know two sides but neither is what I want?
You only need the two sides that appear in whichever ratio matches what you know and what you want. If you are given all three sides, or two sides plus a different unknown, first work out which pair of opposite, adjacent, hypotenuse is relevant to your target, then choose SOH, CAH, or TOA from that pair alone.
Why does finding an angle need shift sin, cos, or tan?
Sin, cos, and tan take an angle and give you a ratio of sides. To go the other way, from a ratio back to an angle, you need the inverse function, written sin⁻¹, cos⁻¹, or tan⁻¹, which is the shift function above the sin, cos, tan keys on your calculator. Work out the ratio first (for example 7.2/5.5), then apply the inverse function to that number.
Does SOH CAH TOA work on any triangle?
No. It only applies to right-angled triangles, and only for angles between 0 and 90 degrees; for trig ratios with obtuse angles, the definitions extend beyond a right-angled triangle entirely. If the triangle in the question does not have a right angle marked, look for one you can construct (for example by dropping a perpendicular), or use the sine rule or cosine rule instead. Most exam word problems that do use these three ratios arrive dressed as angles of elevation and depression, where the right angle comes from the vertical and the horizontal.
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