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}
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