SOH CAH TOA is a memory aid for the three basic trigonometric ratios in a right triangle: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. Name the sides from the angle you are using, choose the ratio that links the sides you know and need, then solve.
In Spanish
SOH CAH TOA (regla mnemotécnica de seno, coseno y tangente)
Label the three sides from the angle θ first. Then each letter group tells you which two sides go in the ratio.
Formal definition and notation
For an acute angleθ in a right triangle, with the hypotenuse across from the right angle, the opposite leg across from θ and the adjacent leg forming θ with the hypotenuse: sin θ = oppositehypotenuse, cos θ = adjacenthypotenuse, tan θ = oppositeadjacent. These ratios depend only on θ, because all right triangles with that angle are similar. SOH CAH TOA is only a mnemonic for these definitions.
Symbol
Read it as
Meaning
SOH
so
sin = oppositehypotenuse
CAH
kuh
cos = adjacenthypotenuse
TOA
toe-uh
tan = oppositeadjacent
Key facts
Find the hypotenuse first: it is across from the right angle. Then opposite and adjacent depend on which acute angle you use.
Switch to the other acute angle and the opposite and adjacent legs trade places.
Sine and cosine of an acute angle are always between 0 and 1; tangent can be any positive number.
Pick the ratio by what you know and need: hypotenuse involved and the opposite leg means SOH, and so on.
Set the calculator to degree mode when angles are in degrees.
The three ratios
The three ratios
sin θ = OH, cos θ = AH, tan θ = OA
O
the leg opposite θ
A
the leg adjacent to θ (not the hypotenuse)
H
the hypotenuse, the longest side
To find an angle, use the inverse key: θ = tan−1(OA), and likewise for sine and cosine.
Try it: sides for the ratios
Interactive
Drag the legs to make a right triangle and read the hypotenuse. With leg a opposite θ, write sin θ = ac, cos θ = bc and tan θ = ab.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
SOH CAH TOA examples, step by step
Example 1: All three ratios
Warm-up
A right triangle has legs 8 and 15 and hypotenuse 17. The angle θ is opposite the side of length 8. Find sin θ, cos θ and tan θ.
Opposite = 8, adjacent = 15, hypotenuse = 17.
SOH: sin θ = 817. CAH: cos θ = 1517.
TOA: tan θ = 815.
Answer:sin θ = 817, cos θ = 1517, tan θ = 815.
Example 2: Find a missing side
Core
In a right triangle, an acute angle is 40° and the leg adjacent to it is 10 cm. Find the leg opposite the angle, to the nearest tenth.
You know the adjacent side and need the opposite side: TOA.
tan 40° = x10, so x = 10 tan 40°.
tan 40° ≈ 0.8391, so x ≈ 8.4.
Answer: The opposite leg is about 8.4 cm.
Example 3: Find an angle
Test-style
A 20-foot ladder leans against a wall and reaches 16 feet up the wall. What angle does the ladder make with the ground, to the nearest tenth of a degree?
The ladder is the hypotenuse, 20. The height, 16, is opposite the angle at the ground.
Opposite and hypotenuse: SOH. sin θ = 1620 = 0.8.
θ = sin−1(0.8) ≈ 53.1°.
Answer: The ladder makes an angle of about 53.1 degrees with the ground.
Common mistakes
✗ MistakeLabeling sides from the wrong angle: using the 8 side as “opposite” for the angle that touches it.
✓ InsteadStand at the angle you are using. The opposite leg is the one it does not touch; the adjacent leg touches it.
Opposite and adjacent swap when you switch to the other acute angle.
✗ MistakeUsing the hypotenuse as the adjacent side in CAH: cos θ = 1715.
✓ Insteadcos θ = adjacent leghypotenuse = 1517.
The hypotenuse touches the angle too, but “adjacent” always means the leg. A sine or cosine above 1 is a sure sign of a mix-up.
✗ MistakeCalculator in radian mode: tan 40 ≈ −1.12, so x ≈ −11.2.
✓ InsteadIn degree mode, tan 40° ≈ 0.839, so x ≈ 8.4.
A negative length or a strange value usually means the wrong angle mode.
✗ MistakeUsing SOH CAH TOA in a triangle with no right angle.
✓ InsteadUse the law of sines or the law of cosines for a triangle without a right angle.
Opposite, adjacent and hypotenuse are defined only for right triangles.
The law of sines works in any triangle, asinA = bsinB. SOH CAH TOA needs a right angle.
Quick check: 3 questions
Question 1 of 3
Which ratio uses the opposite and adjacent sides?
Show answer and explanation
Answer: C. TOA: tangent = opposite over adjacent. Sine uses opposite over hypotenuse, cosine uses adjacent over hypotenuse, and the hypotenuse is a side, not a ratio.
Question 2 of 3
A right triangle has hypotenuse 10 and an acute angle of 30°. How long is the side opposite the 30° angle?
Show answer and explanation
Answer: A. SOH: opposite = 10 sin 30° = 10 × 0.5 = 5. 8.66 uses cosine (the adjacent side), 5.77 uses tangent, and 20 divides by sine.
Question 3 of 3
In a right triangle with sides 7, 24 and 25, angle θ is opposite the side 7. What is cos θ?
Show answer and explanation
Answer: D. CAH: adjacent over hypotenuse = 2425. 725 is sin θ, 247 is the tangent of the other angle, and 2524 is the reciprocal, secant.
SOH CAH TOA: frequently asked questions
What does SOH CAH TOA stand for?
SOH: Sine equals Opposite over Hypotenuse. CAH: Cosine equals Adjacent over Hypotenuse. TOA: Tangent equals Opposite over Adjacent. The letters are the first letters of each word, so saying “so-kuh-toe-uh” reminds you which two sides of a right triangle go in each ratio.
How do I know which ratio to use?
Mark the angle, then label the sides you know and the side you want as opposite, adjacent or hypotenuse. Choose the letter group that has both: opposite and hypotenuse means SOH, adjacent and hypotenuse means CAH, opposite and adjacent means TOA. Then set up the equation and solve.
Does SOH CAH TOA work for any triangle?
No, only for right triangles, because opposite, adjacent and hypotenuse are named from the right angle. For other triangles use the law of sines when you know an angle and its opposite side, or the law of cosines when you know two sides and the angle between them, or all three sides.
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