Right Triangle Trigonometry Cheat Sheet

Key Facts

• For an acute angle $\theta$ in a right triangle, $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.
• For an acute angle $\theta$ in a right triangle, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
• For an acute angle $\theta$ in a right triangle, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.
• Use $\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$, $\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$, or $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$ to find a missing angle.
• The Pythagorean Theorem for every right triangle is $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
• In a $45^\circ-45^\circ-90^\circ$ triangle, the side ratio is $x:x:x\sqrt{2}$.
• In a $30^\circ-60^\circ-90^\circ$ triangle, the side ratio is $x:x\sqrt{3}:2x$, with $x$ opposite $30^\circ$.
• Calculator angle mode matters because degree problems require degree mode, not radian mode.

Vocabulary

Hypotenuse

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

Opposite Side

The opposite side is the side across from the acute angle $\theta$ being used.

Adjacent Side

The adjacent side is the leg next to the acute angle $\theta$ that is not the hypotenuse.

Sine

Sine is the trig ratio $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine

Cosine is the trig ratio $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent

Tangent is the trig ratio $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.