1. Topic: TRIGONOMETRY AND LOGARITHMS
TRIGONOMETRY
Right Triangle
o h
sin θ = csc θ =
h o
a h
cos θ = sec θ =
h a
o a
tan θ = cot θ =
a o
Pythagorean Theorem
h2 = o 2 + a2
COMMON TRIGONOMETRIC IDENTITIES
Reciprocal Identities Sum and Difference of Two Angles
sin θ sin(x + y) = sin x cos y + cos x sin y
tan θ =
cos θ cos(x + y) = cos x cos y + sin x sin y
tan x + tan y
cos θ tan(x + y) =
cot θ = 1 − tan x tan y
sin θ
If considering the difference, just change
1 sign to negative.
cos θ =
sec θ Double Angle Identities
sin θ
sin θ = sin 2x = 2 sin x cos x
csc θ
Pythagorean Relations cos 2x = cos 2 x − sin2 x
2 tan x
tan 2x =
sin2 θ + cos 2 θ = 1 1 − tan2 x
tan2 θ + 1 = sec 2 θ
1 + cot 2 θ = csc 2 θ
Sine and Cosine Law
a b c
= =
sin A sin B sin C
a2 = b2 + c2 − 2bc cos A
LOGARITHMS
Properties of Logarithms
loga M = N means aN = M
Identity Illustration
log3 21 = log3 7 + log3 3
loga MN = loga M + loga N
M 7
loga = loga M − loga N log2 = log2 7 − log2 3
N 3
loga (M) = N loga M log2 (7) = 3 log2 7
N 3
loga a = 1 log3 3 = 1
loga 1 = 0 log3 1 = 0
logb r log2 3
b =r 2 =3
log b log 5
loga b = log2 5 = = 2.3219
log a log 2
Natural Logarithms
Natural logarithms are logarithms to the base e, where e (Naperian base) is approximately equal to 2.71828
ln e = 1
ln e n = n
e ln x
=x
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