Key takeaway: SOHCAHTOA is a three-chunk mnemonic for right triangles: sine pairs the opposite side with the hypotenuse, cosine pairs the adjacent side with the hypotenuse, and tangent pairs the two legs.
Grade 10 – GeometryG-SRT.C.6Trigonometry
Definition of SOHCAHTOA
SOHCAHTOA squeezes three definitions into one odd word. Read it in three chunks: SOH, CAH and TOA. Each chunk names a trig ratio, then the two sides it divides.
Which leg is opposite and which is adjacent depends on the acute corner you choose. The leg across from the corner is opposite, the leg touching it is adjacent, and the hypotenuse always faces the square corner. Switch to the other acute corner and the two legs trade jobs. That is why SOHCAHTOA starts with labeling.
Formula and notation
Name the opposite leg \(o\), the adjacent leg \(d\) and the hypotenuse \(h\), all seen from an acute angle \(\alpha\). SOHCAHTOA then says:
\[ \sin\alpha = \frac{o}{h}, \qquad \cos\alpha = \frac{d}{h}, \qquad \tan\alpha = \frac{o}{d} \]
Label first, then pick the ratio containing the two sides in your problem.
Key facts
- Label the sides from the viewpoint of the angle you are using.
- The hypotenuse stays the longest side whichever angle is chosen.
- Sine and cosine of an acute angle always fall between 0 and 1.
- A calculator must be set to degree mode when angles are in degrees.
SOHCAHTOA example
An 18-ft extension ladder leans on a barn wall with its foot on level ground, tilted \(64^{\circ}\) up from the ground. How high on the barn does the top touch, and how far from the barn is the base? Use \(\sin 64^{\circ} \approx 0.8988\) and \(\cos 64^{\circ} \approx 0.4384\).
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The ladder is the hypotenuse (18 ft). The height is across from the \(64^{\circ}\) corner, so SOH applies: \(\sin 64^{\circ} = \frac{h}{18}\).
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Height: \(h = 18 \times 0.8988 \approx 16.2\) ft.
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The base distance touches the angle, so CAH applies: \(d = 18 \times 0.4384 \approx 7.9\) ft.
The top touches the barn about 16.2 ft up and the base sits about 7.9 ft away.
Common mistake
Watch out
For the same ladder, Dev writes \(\cos 64^{\circ} = \frac{h}{18}\), reasoning that the height is “the up one” and so belongs with cosine.
Why it is wrong
Direction does not decide the ratio; position relative to the angle does. The height is across from the \(64^{\circ}\) corner, so it is the opposite side and goes with sine. The distance touching the corner goes with cosine.
Related terms and lessons
- Sine
- Cosine
- Tangent (Trigonometric Ratio)
- Pythagorean Theorem
- Hypotenuse
- Angle of Elevation and Depression
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