Trigonometric Identities And Formulas

Trigonometric Identities and Formulas
Below are some of the most important definitions, identities and formulas in trigonometry.
1. Trigonometric Functions of Acute Angles
sin X = opp / hyp = a / c, csc X = hyp / opp = c / a

tan X = opp / adj = a / b , cot X = adj / opp = b / a

cos X = adj / hyp = b / c , sec X = hyp / adj = c / b ,

2. Trigonometric Functions of Arbitrary Angles
sin X = b/ r , csc X = r / b

tan X = b / a , cot X = a / b

cos X = a / r , sec X = r / a

3. Special Triangles
Special triangles may be used to find trigonometric functions of special angles: 30, 45 and60 degress.


4. Sine and Cosine Laws in Triangles
In any triangle we have:

1 - The sine law

sin A / a = sin B / b = sin C / c

2 - The cosine laws

a 2 = b 2 + c 2 - 2 b c cos A

b 2 = a 2 +c 2 - 2 a c cos B

c 2 = a 2 + b 2 - 2 a b cos C

5. Relations Between Trigonometric Functions
cscX = 1 / sinX

sinX = 1 / cscX

secX = 1 / cosX

cosX = 1 / secX

tanX = 1 / cotX

cotX = 1 /tanX

tanX = sinX / cosX

cotX = cosX / sinX
6. Pythagorean Identities
sin 2X + cos 2X = 1

1 + tan 2X = sec 2X

1 + cot 2X = csc 2X

7. Negative Angle Identities
sin(-X) = - sinX , odd functioncsc(-X) = - cscX , odd function

cos(-X) = cosX , even function

sec(-X) = secX , even function

tan(-X) = - tanX , odd function

cot(-X) = - cotX , odd function
8. Cofunctions Identitiessin(pi/2 - X) = cosX

cos(pi/2 - X) = sinX

tan(pi/2 - X) = cotX

cot(pi/2 - X) = tanX

sec(pi/2 - X) = cscX

csc(pi/2 - X) = secX
9. Addition Formulas
cos(X + Y) = cosX cosY - sinX sinY

cos(X - Y)= cosX cosY + sinX sinY

sin(X + Y) = sinX cosY + cosX sinY

sin(X - Y) = sinX cosY - cosX sinY

tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY]

tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY]cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY]

cot(X - Y) = [ cotX cotY + 1 ] / [ cotY - cotX]
10. Sum to Product Formulas
cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]

sinX + sinY =...