Trigonometric Identity
TrigonometryA trigonometric identity is an equation involving trigonometric functions that is true for all valid values of the variable.
Definition
A trigonometric identity is a math equation involving sine, cosine, or tangent that is always true, no matter what angle you use.
Example
$\sin^2(\theta) + \cos^2(\theta) = 1$ is a trig identity. Plug in any angle, say $30^\circ$: $(0.5)^2 + (\sqrt{3}/2)^2 = 0.25 + 0.75 = 1$. Always true.
Key Insight
Identities are like shortcuts. Instead of doing complicated trig calculations from scratch, you can swap one expression for an equal one to simplify your work.
Definition
A trigonometric identity is an equation that holds for all values of the variable for which both sides are defined. Major categories include Pythagorean identities, reciprocal identities, quotient identities, and co-function identities (e.g., $\sin(90^\circ - \theta) = \cos(\theta)$).
Example
Prove: $\tan(\theta)\cos(\theta) = \sin(\theta)$. Left side $= (\sin/\cos)\cos = \sin$. So LHS $=$ RHS for all $\theta$ where $\cos(\theta) \neq 0$. This is a proof by substitution using the quotient identity.
Key Insight
Trig identities are used to simplify integrals in calculus, solve trig equations, and transform expressions. Recognizing which identity to apply is a key skill that comes with practice.
Definition
A trigonometric identity is a universally valid equation in the algebra of trigonometric functions. They arise from the properties of the exponential function via Euler's formula: $e^{ix} = \cos(x) + i\sin(x)$. The sum formulas $\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ follow from $e^{i(a+b)} = e^{ia}e^{ib}$, making complex multiplication the generating mechanism for all angle-addition identities.
Example
The double-angle identity $\cos(2x) = \cos^2(x) - \sin^2(x)$ follows from $e^{2ix} = (e^{ix})^2 = (\cos(x) + i\sin(x))^2 = \cos^2(x) - \sin^2(x) + 2i\sin(x)\cos(x)$, taking the real part.
Key Insight
Every trigonometric identity can be derived from two facts: Euler's formula $e^{ix} = \cos(x) + i\sin(x)$, and properties of the exponential function ($e^{a+b} = e^a e^b$). This reduces the entire algebra of trig identities to complex exponential arithmetic.