SOH-CAH-TOA: the mnemonic that locks in sin, cos, tan in 30 seconds

The mnemonic

In a right triangle, for an angle $\theta$ (not the right angle):

Mnemonic	Meaning
SOH	Sin θ = Opposite / Hypotenuse
CAH	Cos θ = Adjacent / Hypotenuse
TOA	Tan θ = Opposite / Adjacent

Identify the 3 sides

Relative to angle $\theta$ (not the right angle):

• Opposite: the side across from the angle
• Adjacent: the side next to the angle (but not the hypotenuse)
• Hypotenuse: the longest, always opposite the right angle

Example

Right triangle with angle $\theta$, opposite = 3 cm, adjacent = 4 cm.

• $\tan \theta = 3/4 = 0.75$
• Hypotenuse: Pythagoras → $\sqrt{9 + 16} = 5$
• $\sin \theta = 3/5 = 0.6$
• $\cos \theta = 4/5 = 0.8$

Find an angle

Given 2 sides, use inverse trig:

• $\sin^{-1}(0.6)$ (or “asin”) → about $36.87°$.

Sanity checks

• Angle between 0° and 90° (right triangle).
• sin and cos between 0 and 1.
• tan can be > 1 (when opposite > adjacent).

Common mistakes

1. Mixing opposite and adjacent — always relative to your angle.
2. Using the right angle as reference — no, use the other acute angle.
3. Degree vs radian mode on the calculator — in middle school, ALWAYS degrees.

?À toi de jouer

In a right triangle, one angle is 30°. The opposite side is 5 cm. Find the hypotenuse.

A.10 cmB.5√3 cmC.√3 cmD.5/2 cm

Valider