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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