Pre-Lecture 3: Precalculus Trigonometry (Sections 1.2, 1.3 & 1.5 ...
[Pages:13]Pre-Lecture 3: Precalculus Trigonometry (Sections 1.2, 1.3 & 1.5)
Trigonometric Functions 1. Trigonometric functions
hypotenus
adjacent
opposite
P(x,y)
sin =
csc =
cos =
sec =
tan =
cot =
In general, let P (x, y) be any point on the terminal side of (radians) and let r = x2 + y2 be the distance from the origin to point P .
L3 - 2
2. Unit circle (r = 1, so sin = y and cos = x)
2/3 3/4 5/6
/2 /3 /4 /6
0
7/6
5/4 4/3 3/2
11/6
7/4 5/3
0
/6
/4
/3
/2
sin
cos
tan
3. Graphs y = sin x
L3 - 3
y = cos x
y = tan x
y = cot x
y = sec x NOTE: sin x
y = csc x , cos x
L3 - 4
ex. Sketch the graph of y = -2 sin 21x .
1
-
-1
Basic Trigonometric Identities
1) sin2 + cos2 = 1 2) tan2 + 1 = sec2 3) 1 + cot2 = csc2
4) sin(-) = - sin 5) cos(-) = cos
L3 - 5
Addition/Subtraction Formulas 6) sin(x ? y) = sin x cos y ? cos x sin y
7) cos(x ? y) = cos x cos y sin x sin y
8)
tan(x
?
y)
=
tan x ? tan y 1 tan x tan y
Double-Angle Formulas 9) sin(2x) = 2 sin x cos x
10) cos(2x) = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x
Half-Angle Formulas 11) cos2 x = 1 + cos(2x)
2
12) sin2 x = 1 - cos(2x) 2
L3 - 6
ex.
If
is
in
the
third
quadrant
with
tan
=
2 3
and
is
in the first quadrant with sin = 35, find the exact value of
sec( + ).
Inverse Trigonometric Functions ? y = sin-1 x if and only if
y = sin x
1
-
-1
/2
-1
1
-/2
? y = cos-1 x if and only if
y = cos x
1
-
-1
/2
-1
1
? y = tan-1 x if and only if
L3 - 7
y = tan x
1
-
-1
/2
-1
1
-/2
There are similar definitions for the inverse of the other trigonometric functions.
L3 - 8
ex. Find the following if possible:
1) sin-1
1 -
2
2) cos-1(2) 3) tan-1 tan 3
4
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