Trigonometric Identities & Formulas

Right Triangle Definitions of Trigonometric Functions

Trigonometric Values of Special Angles

Reciprocal Identities:

\[ sin x = \frac{1}{csc x} \: \: \: \: \: \: \: \: \: \: \: \:    csc x = \frac{1}{sin x} \]

\[ cos x = \frac{1}{sec x} \: \: \: \: \: \: \: \: \: \: \: \:    sec x = \frac{1}{cos x} \]

\[ tan x = \frac{1}{cot x} \: \: \: \: \: \: \: \: \: \: \: \:    cot x = \frac{1}{tan x} \]

\[ tan x = \frac{sin x}{cos x} \: \: \: \: \: \: \: \: \: \: \: \:    cot x = \frac{cos x}{sin x} \]

Pythagorean Identities

\[sin^2 x + cos^2 x = 1\]

\[ sec^2 x - tan^2 x = 1\]

\[ csc^2 x - cot^2 x = 1\]

Odd-Even Identities (Negative angle identities)

sin (-x) = - sin(x)                                        csc(-x) = - csc(x)

cos (-x) =  cos(x)                                        sec(-x) =  sec(x)

tan(-x) = - tan(x)                                        cot (-x) = - cot(x)

Sum and Difference Formulas

sin(A+B) = sinAcosB + cosAsinB               sin(A-B) = sinAcosB - cosAsinB

cos(A+B) = cosAcosB - sinAsinB                cos(A-B) = cosAcosB + sinAsinB

\[ tan(A+B) = \frac{tanA + tanB}{1 - tanAtanB} \: \: \: \: \: \: \: \: \: \: \: \:tan(A-B) = \frac{tanA - tanB}{1 + tanAtanB} \]

Sum-to-Product Formulas

\[sinC + sinD = 2sin(\frac{C+D}{2})cos(\frac{C-D}{2}) \]

\[sinC - sinD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) \]

\[cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2}) \]

\[cosC - cosD = - 2sin(\frac{C+D}{2})sin(\frac{C-D}{2}) \]

Other Basic Identities

\[sin2A = 2sinAcosA\]

\[cos2A = cos^A - sin^2A = 2cos^2A -1 = 1 - 2sin^2A\]

\[tan2A = \frac{2tanA}{1- tan^2A}\]

\[sin^2A = \frac{1 - cos2A}{2}\]

\[cos^2A = \frac{1 + cos2A}{2}\]

Law of Sines and Cosines

\[\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}\]

\[cosA = \frac{b^2 + c^2 -a^2}{2bc}\]

\[cosB = \frac{a^2 + c^2 - b^2}{2ac}\]

\[cosC = \frac{a^2 + b^2 - c^2}{2ab}\]

Area of an Oblique Triangle

\( Area = \frac{1}{2}bc sinA = \frac{1}{2}ac sinB = \frac{1}{2}ab sinC\)

Heron’s Formula:

\[Area = \sqrt{s(s-a)(s-b)(s-c)} \]

where $s = \frac{a+b+c}{2}$