Mathematics › 🔍 Grade 10 › Right-Triangle Trigonometry (SOHCAHTOA)

Right-Triangle Trigonometry (SOHCAHTOA)

Name a triangle's sides relative to one acute angle, and three fixed ratios unlock every unknown side or angle.

Grade 10Geometry

Right-Triangle Trigonometry (SOHCAHTOA) — illustration

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The big idea: In any right triangle, once you pick an acute angle θ to measure from, the other two sides earn names — opposite and adjacent — relative to that angle. The ratios opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent (sine, cosine and tangent) are well-defined numbers because, as the last lesson showed, every right triangle with the same angle θ is similar. These three ratios let you find an unknown side from an angle, or an unknown angle from two sides.

🎯 By the end, you'll be able to

Label a right triangle's sides as opposite, adjacent and hypotenuse relative to a chosen acute angle
Apply sine, cosine and tangent (SOHCAHTOA) to find an unknown side
Use inverse trig functions to find an unknown angle from two known sides
Recall the exact trig values for 30°, 45° and 60° from the half-equilateral and half-square triangles

📎 You should already know

Triangle similarity criteria (previous lesson)
The Pythagorean theorem

Why these ratios are well-defined

The last lesson proved something that makes this one possible: any two right triangles that share an acute angle θ are similar (by AA), so their sides are proportional. That means the ratio of any two sides — say, the side opposite θ to the hypotenuse — is the same number in every right triangle with that angle, no matter its size. That fixed ratio is what we're about to name.

🔑 Naming the sides — and SOHCAHTOA

Pick an acute angle θ in a right triangle. The hypotenuse is the side opposite the right angle (always the longest). The opposite side is across from θ; the adjacent side touches θ but isn't the hypotenuse. Then: SOH (sin θ = opp⁄hyp), CAH (cos θ = adj⁄hyp), TOA (tan θ = opp⁄adj).

\[ \sin\theta=\frac{\text{opp}}{\text{hyp}} \qquad \cos\theta=\frac{\text{adj}}{\text{hyp}} \qquad \tan\theta=\frac{\text{opp}}{\text{adj}} \]

The three basic trigonometric ratios, defined for an acute angle θ in a right triangle.

⚠️ Opposite and adjacent swap; the hypotenuse never does

‘Opposite’ and ‘adjacent’ are defined relative to whichever acute angle you call θ. Switch to the other acute angle, and the side that was opposite becomes adjacent, and vice versa. The hypotenuse never changes — it is always the side opposite the right angle and always the longest side, no matter which acute angle you're working from.

🎮 Right-Triangle Solver LIVE

Set an acute angle and one side, then watch SOHCAHTOA solve for the rest of the triangle.

📝 Worked example: In a right triangle, θ = 35° and the hypotenuse is 10. Find the side opposite θ.

SOH says sin θ = opp⁄hyp, so opp = hyp × sin θ.
Substitute: opp = 10 × sin(35°).
sin(35°) ≈ 0.574, so opp ≈ 10 × 0.574.

✓ The opposite side is about <strong>5.74</strong> units.

Working backwards: inverse trig

If you know two sides but not the angle, run SOHCAHTOA in reverse with the inverse trig functions sin⁻¹, cos⁻¹ and tan⁻¹. If tan θ = opp⁄adj, then θ = tan⁻¹(opp⁄adj). These are the everyday tools behind the angle of elevation (looking up from horizontal) and angle of depression (looking down from horizontal).

📝 Worked example: A ramp rises 3 ft over a horizontal run of 15 ft. Find the ramp's angle of elevation θ.

The rise (3 ft) is opposite θ; the run (15 ft) is adjacent to θ. That's TOA: tan θ = opp⁄adj.
tan θ = 3⁄15 = 0.2.
θ = tan⁻¹(0.2).

✓ θ ≈ <strong>11.3°</strong>.

✨ Co- means complementary

The two acute angles of a right triangle always add to 90° — they're complementary. Swap which one you call θ and opposite/adjacent swap too, which means sin θ = cos(90° − θ). That's the origin of the prefix ‘co’ in cosine: it's the sine of the complementary angle.

Exact values from two special triangles

Cut an equilateral triangle in half and you get a 30–60–90 triangle with sides in ratio 1 : √3 : 2. Cut a square along its diagonal and you get a 45–45–90 triangle with sides in ratio 1 : 1 : √2. Because these ratios never change (similarity again), they give exact trig values instead of decimal approximations.

\[ \sin30^{\circ}=\tfrac12,\ \cos30^{\circ}=\tfrac{\sqrt3}{2},\ \tan30^{\circ}=\tfrac{1}{\sqrt3}\qquad \sin45^{\circ}=\cos45^{\circ}=\tfrac{\sqrt2}{2},\ \tan45^{\circ}=1\qquad \sin60^{\circ}=\tfrac{\sqrt3}{2},\ \cos60^{\circ}=\tfrac12,\ \tan60^{\circ}=\sqrt3 \]

Exact sine, cosine and tangent values for 30°, 45° and 60°, read directly off the half-equilateral and half-square triangles.

⚠️ tan θ has no ceiling — and no value at 90°

As θ approaches 90°, the adjacent side shrinks toward zero while tan θ = opp⁄adj grows without bound. At exactly 90° there is no right triangle left to measure — tan(90°) is undefined. Sine and cosine have no such problem: both stay between −1 and 1 for every angle.

Where this goes next

SOHCAHTOA only works for acute angles inside a right triangle — there's no ‘opposite side’ for a 120° angle, since 120° can't fit inside a right triangle at all. The next lesson frees sine and cosine from the triangle entirely, redefining them using the unit circle so they make sense for any angle.

Check your understanding

1. For an acute angle θ in a right triangle, sin θ equals:

SOH: sine is the opposite side divided by the hypotenuse.

2. Which side of a right triangle never changes, no matter which acute angle you call θ?

The hypotenuse is always the side opposite the right angle and the longest side; only the labels 'opposite' and 'adjacent' swap when you change which acute angle is θ.

3. A right triangle has opposite side 7 and adjacent side 10 relative to angle θ. Which expression finds θ?

tan θ = opposite/adjacent = 7/10, so solving for the angle itself requires the inverse tangent: θ = tan⁻¹(7/10).

4. Why does sin θ = cos(90° − θ)?

The two acute angles sum to 90°. Naming the other one (90° − θ) swaps which side counts as opposite and which as adjacent, turning sin θ into cos(90° − θ) — the origin of the 'co' prefix.

5. What happens to tan θ as θ approaches 90°?

tan θ = opposite/adjacent, and the adjacent side shrinks toward zero as θ nears 90°, sending the ratio toward infinity; at 90° there is no right triangle left, so tan(90°) is undefined.

✅ Key takeaways

Opposite, adjacent and hypotenuse are named relative to a chosen acute angle θ; the hypotenuse never changes, but opposite/adjacent swap if you switch θ.
SOHCAHTOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj — well-defined because similar right triangles share these ratios.
Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) work backwards from two known sides to find an unknown angle, including angles of elevation and depression.
The two acute angles are complementary, giving sin θ = cos(90° − θ); tan θ grows without bound as θ → 90° and is undefined there.
The half-equilateral (30-60-90) and half-square (45-45-90) triangles give exact trig values for 30°, 45° and 60°.