Right Triangle Trigonometry SOHCAHTOA Cheat Sheet

Key Facts

• In a right triangle, the hypotenuse is always the side opposite the $90^\circ$ angle and is the longest side.
• For an acute angle $\theta$, $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$.
• For an acute angle $\theta$, $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.
• For an acute angle $\theta$, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.
• SOHCAHTOA means $\sin=\frac{O}{H}$, $\cos=\frac{A}{H}$, and $\tan=\frac{O}{A}$.
• To solve for a missing side, set up the correct ratio, substitute known values, then use algebra to isolate the variable.
• To solve for a missing angle, use inverse trig such as $\theta=\sin^{-1}\left(\frac{O}{H}\right)$, $\theta=\cos^{-1}\left(\frac{A}{H}\right)$, or $\theta=\tan^{-1}\left(\frac{O}{A}\right)$.
• Trigonometric ratios depend on the chosen angle, so the opposite and adjacent sides can change when a different acute angle is used.

Vocabulary

Right triangle

A triangle with one angle measuring exactly $90^\circ$.

Hypotenuse

The side opposite the $90^\circ$ angle and the longest side of a right triangle.

Opposite side

The side across from the acute angle being used in a trigonometric ratio.

Adjacent side

The leg next to the acute angle being used, not including the hypotenuse.

Sine

A trig ratio defined by $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$ in a right triangle.

Inverse trigonometric function

A function such as $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ used to find an angle from a side ratio.