Math 140
Lecture 23
RECALL
cosx  y  cos x cos y  sin x sin y
cosx  y  cos x cos y  sin x sin y
Adding these gives
cosx  y  cosx  y  2 cos x cos y
Subtracting gives
cosx  y  cosx  y  2 sin x sin y
Similarly
sinx  y  sin x cos y  cos x sin y
sinx  y  sin x cos y  cos x sin y
Adding these gives
sinx  y  sinx  y  2 sin x cos y
Dividing by 2 gives
PRODUCT-TO-SUM FORMULAS
sin x sin y  12 [cosx  y  cosx  y ]
cos x cos y  12 [cosx  y  cosx  y ]
sin x cos y  12 [sinx  y  sinx  y ]  cos y sin x
sin x sin y  12 [cosx  y  cosx  y ]  sin y sin x
cos x cos y  12 [cosx  y  cosx  y ]  cos y cos x
sin x cos y 
__
1
[
]
2 sinx__ y  sinx  y
In the last formula,
x is the argument of sin,
y is the argument of cos.
`cos a sin b  ?
`sin b sin a  ?
Careful.
 cos y sin x
__
sin x sin y  12 [cosx  y  cosx  y ]
cos x cos y  12 [cosx  y  cosx  y ]
sin x cos y  12 [sinx  y  sinx  y ]  cos y sin x
Write as a sum or difference of trigonometric functions.
`sin 2
cos
`cos 6

4
.
.
.
.
sin

2
sin x sin y  12 [cosx  y  cosx  y ]
cos x cos y  12 [cosx  y  cosx  y ]
sin x cos y  12 [sinx  y  sinx  y ]  cos y sin x
Write as a sum or difference of trigonometric functions.
`cos x cos 2x
`sinx  ysinx  y
.
.
Solving trigonometric equations
cos x  cosx, if x is one solution, -x is another solution.
sin x  sin  x if x is one solution, p- x is another.
`sin   12 . Find all solutions.
y= 1/2
p-p/6=5p/6
p/6
Two simplest solutions:
q = p/6,
q = p - p/6 = 5p/6.
sin(q) has period 2p,
tadding multiples of 2p to a solution is also a solution.
General solution: two sets
q = p/6 + 2pn, q = 5p/6 + 2pn
b `cos   12 . Find all solutions.
To solve sin  34 , the simplest solution on the right is
  sin1 34 , the solution on the left is     sin1 34 .
To solve cos  34 , the simplest solution above the x-axis
is   cos 1  34 , the solution below    cos 1  34 .
CONVENTION. n is an arbitrary, possibly negative, integer.
For sin, cos, add 2pn to the (usually two) simplest solutions.
For tan, add pn to the one simplest solution.
Find all solutions.
`tan x  1
`tan x  1
3
... /6  n
For sin, cos, add 2pn to the (usually two) simplest solutions.
For tan, add pn to the one simplest solution.
Find all solutions.
`cos x  3
`sin 2x  1
`cos 2x  1
.
.
... /4  n
For sin, cos, add 2pn to the (usually two) simplest solutions
For tan, add pn to the one simplest solution.
cos   cos, if  is one solution, - is another solution.
sin   sin   if  is one solution, p-  is another.
sin(p/6) = 1/ 2
p/2
sin(p/4) = 1/ 2
sin(p/3) = l3̄/2
cos(p/6) =
3 /2
cos(p/4) = 1/ 2
cos(p/3) = 1/2
p-q
q
0
p
-q
tan(p/6) = 1/l3̄
tan(p/4) = 1
tan(p/3) = l3̄
Remember these values.
-p/2
Find all solutions. Three sets.
`2 cos 2 x  cos x  1
``2 sin 2 x  sin x  1
.
.
.
.
.
.
.
Find all solutions.
Recall: sin 2 x  cos 2 x  1
`2 cos 2 x  sin x  1  0
Rewrite entirely in sin or entirely in cos.
Find all solutions. Recall: sin 2 x  cos 2 x  1
`sin 2 x  cos x  1  0
``cos 2 x  sin x  1  0
.
.
.
.
.
.
`2 tan 2 x  3 tan x sec x  2 sec 2 x  0
b
sin, cos and tan are not 1-1. The x-axis is a horizontal line
which crosses their graphs more than once.
They are 1-1 when restricted to the green intervals.
sin
-p/2
p/2
cos
p
0
tan
-p/2
For sin,
For cos,
For tan,
p/2
the restricted interval is [-p/2, p/2].
the restricted interval is [0, p].
the restricted interval is (-p/2, p/2).
Inverse trigonometric functions
sin -1(x) = the q i [-p/2, p/2] such that sin(q) = x,
cos -1(x) = the q i [0, p]
such that cos(q) = x,
tan -1(x) = the q i (-p/2, p/2) such that tan(q) = x.
sin-1(x) and cos -1(x) have domain [-1, 1].
tan -1(x) has domain (-5, 5).
NOTATION
arcsin(x) = sin -1(x)
arccos(x) = cos -1(x)
arctan(x) = tan -1(x)
WARNING. sin -1(x) = 1/sin(x).
sin -1(x) = arcsin x = the inverse
(sin(x))
bb
-1
1
= sin x = the reciprocal