A right triangle has an opposite side of 3 and a hypotenuse of 5. What is tan θ? Most people answer 4/5 — and most people are wrong. The answer is 3/4. The trap is grabbing the wrong side as the "adjacent," and it is the single most common trigonometry mistake in school math. Here is the trap, the fix, and the three ratios that solve any right triangle.
The 3/5 trap: tan θ is 3/4, not 4/5
Start with the three ratios, memorized as SOHCAHTOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. The ratios are about which side sits where relative to the angle — and the adjacent side is the one students misidentify.
We know sin θ = 3/5, so the opposite side is 3 and the hypotenuse is 5. The adjacent side is not given — find it with the Pythagorean theorem: adjacent² = 5² − 3² = 25 − 9 = 16, so adjacent = 4. Now tan θ = opposite ÷ adjacent = 3 ÷ 4 = 0.75, or 3/4.
The wrong answer, 4/5, comes from dividing the adjacent (4) by the hypotenuse (5) — which is cos θ, not tan θ. The rule worth memorizing: tan uses opposite over adjacent — the hypotenuse never appears in tan. If your tan answer has the hypotenuse in it, you have computed a different ratio.
Check it a second way, with the angle itself. The inverse sine finds the angle from the ratio: θ = arcsin(3/5) ≈ 36.87°. Then tan(36.87°) ≈ 0.75 — the same 3/4, two completely different routes. And cos θ = 4/5 = 0.8 confirms the side we found: adjacent = 4, hypotenuse = 5. The wrong answer 4/5 is cos θ wearing tan's label.
The three ratios, and the angles worth knowing by heart
Sin, cos, tan — what each one is. Sin compares the opposite side to the hypotenuse, cos compares the adjacent side to the hypotenuse, and tan compares the opposite to the adjacent. Sin and cos always have the hypotenuse underneath; tan never does.
The three angles to memorize. At 30°: sin 30° = 0.5, cos 30° ≈ 0.8660, tan 30° ≈ 0.5774. At 45°: sin 45° = cos 45° ≈ 0.7071, tan 45° = 1 — the only angle where sin and cos are equal, which is why tan is exactly 1. At 60°: sin 60° ≈ 0.8660, cos 60° = 0.5, tan 60° ≈ 1.7321. Notice the mirror: sin 30° equals cos 60° (both 0.5), and sin 60° equals cos 30° (both 0.8660). That is not a coincidence — sin of an angle is cos of its complement, because the two acute angles in a right triangle add to 90°.
Inverse trig: finding the angle
Trigonometry also runs backwards. If you know the sides, the inverse functions — arcsin, arccos, arctan — find the angle. The 3-4-5 triangle again: θ = arcsin(3/5) = arccos(4/5) = arctan(3/4) ≈ 36.87°, all three giving the same angle, which is the built-in check that the sides are consistent. The other acute angle is 90° − 36.87° = 53.13°, and the three angles add to exactly 180°.
The simplest inverse example is the one every student should know cold: sin θ = 0.5 means θ = 30°, because sin 30° = 0.5. And tan θ = 1 means θ = 45°, because tan 45° = 1.
The right-triangle solver: finding a missing side
Given one side and one acute angle, the ratios find everything else. Example: hypotenuse 10, angle 30°. Opposite = 10 × sin(30°) = 10 × 0.5 = 5. Adjacent = 10 × cos(30°) = 10 × 0.8660 ≈ 8.66. Check with Pythagoras: 5² + 8.66² = 25 + 75 = 100 = 10² — the triangle closes, so the sides are right.
Two more triangles worth knowing by heart. The 5-12-13 triangle: the angles are arcsin(5/13) ≈ 22.62° and arcsin(12/13) ≈ 67.38°.
The checks that catch every error
Four habits prevent nearly all trigonometry mistakes. First, name the sides before you compute — opposite, adjacent, hypotenuse, relative to your angle. The side that is opposite one angle is adjacent to the other, and most errors start here. Second, the angles must add to 180° — in the 3-4-5 triangle, 36.87° + 53.13° + 90° = 180° exactly. Third, check the mode: a calculator in radian mode returns sin(30) = −0.988, not 0.5 — check degrees before trusting any number.
Check it with code
The whole topic in one Python block:
import math
# 3/5 trap: adjacent first, then tan
print(math.sqrt(5**2 - 3**2)) # 4.0
print(3/4) # 0.75 (NOT 4/5 = 0.8)
print(math.degrees(math.asin(3/5))) # 36.8699...
print(math.tan(math.radians(36.86989765))) # 0.75
# The three angles to memorize
print(math.sin(math.radians(30))) # 0.5
print(math.tan(math.radians(45))) # 1.0
print(math.cos(math.radians(30))) # 0.8660254
# Solver: hyp 10, angle 30 -> opp 5, adj 8.66
print(10 * math.sin(math.radians(30))) # 5.0
print(10 * math.cos(math.radians(30))) # 8.660254...
print(5**2 + 8.660254037844387**2) # 100.0
A quick drill
- sin θ = 3/5, hypotenuse 5: adjacent = 4, tan θ = 3/4 — never 4/5, because 4/5 is cos θ.
- sin 30° = 0.5; tan 45° = 1; cos 60° = 0.5.
- θ = arcsin(3/5) ≈ 36.87°; the other acute angle is 53.13°.
- Hypotenuse 10, angle 30°: opposite = 5, adjacent ≈ 8.66, and 5² + 8.66² = 100.
- 5-12-13 triangle: angles ≈ 22.62° and 67.38°.
Want the steps shown instead of just the answers? A free trigonometry calculator solves all three modes — sin, cos, tan from an angle or from the sides, inverse trig to find the angle, and the full right-triangle solver — with every step worked out. If the ratios still feel slippery, its guide to how to find sin cos tan walks through SOHCAHTOA with the 3/5 trap as the worked example.
One rule, three ratios: name the sides first, remember tan never uses the hypotenuse, and let the inverse functions check your angle. Check your own triangles on a free trigonometry calculator and the trap stops working on you.
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