Trigonometric Ratios and Functions

When working with right-angled triangles, we need to identify three key sides: the hypotenuse (longest side, opposite the right angle), the opposite side (across from the angle we're working with), and the adjacent side (next to our angle).

Pythagoras's Theorem helps us find any missing side:

• $a^2 = b^2 - c^2$ (Adjacent)
• $b^2 = a^2 + c^2$ (Hypotenuse)
• $c^2 = b^2 - a^2$ (Opposite)

The most important trigonometric ratios can be remembered using the memory aid SOHCAHTOA:

• Sine: $\sin\theta = \frac{Opposite}{Hypotenuse} = \frac{c}{b}$
• Cosine: $\cos\theta = \frac{Adjacent}{Hypotenuse} = \frac{a}{b}$
• Tangent: $\tan\theta = \frac{Opposite}{Adjacent} = \frac{c}{a}$

Pro Tip: The other three ratios (cosecant, secant, and cotangent) are just the reciprocals of sine, cosine, and tangent. For example, $\cot\theta = \frac{1}{\tan\theta} = \frac{Adjacent}{Opposite}$