Simple Trigonometric Identities

The Tangent Identity defines the relationship between the three primary trigonometric ratios: $\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$

This identity is derived from the right-angled triangle definitions where $\sin \theta = \frac{O}{H}$, $\cos \theta = \frac{A}{H}$, and $\tan \theta = \frac{O}{A}$. Dividing sine by cosine cancels the hypotenuse, leaving the ratio of opposite to adjacent.

It is particularly useful for converting equations containing both sine and cosine into a single trigonometric function (tangent), which is often easier to solve.

The Pythagorean Identity is the most fundamental relationship in trigonometry: $\sin^2 \theta + \cos^2 \theta \equiv 1$

This principle is a direct application of Pythagoras' Theorem ($a^2 + b^2 = c^2$) to a unit circle, where the radius (hypotenuse) is 1, the vertical rise is $\sin \theta$, and the horizontal run is $\cos \theta$.

This identity can be rearranged to isolate either squared term: $\sin^2 \theta \equiv 1 - \cos^2 \theta$ or $\cos^2 \theta \equiv 1 - \sin^2 \theta$.

Note that $\sin^2 \theta$ is a shorthand notation for $(\sin \theta)^2$; it does not mean $\sin(\theta^2)$.

Substitution: When an equation contains multiple trigonometric functions, use identities to replace one function with another to achieve uniformity (e.g., replacing $\cos^2 \theta$ with $1 - \sin^2 \theta$ in a quadratic equation).

Simplification: Use the tangent identity to reduce fractions involving sine and cosine into a single term, or vice versa, to align with other terms in the expression.

Verification: To prove a new identity, start with the more complex side and apply known identities step-by-step until it matches the simpler side.

Identify the 'Odd One Out': If an equation is mostly in terms of sine but has one $\cos^2$ term, that $\cos^2$ term is your target for substitution using the Pythagorean identity.

Show Every Step: Examiners award marks for the explicit substitution of identities. Never skip the line where you replace a term with its equivalent identity.

Check the Power: The Pythagorean identity only applies to squared terms. You cannot directly replace $\sin \theta$ with $1 - \cos \theta$; you must have $\sin^2 \theta$.

Tangent Conversion: If you see an equation like $a \sin \theta = b \cos \theta$, your first instinct should be to divide by $\cos \theta$ to create a $\tan \theta$ term.