T r i g o n o m e t r y T r i v i u m
[Pages:5]Trigonometry Trivium
Main Trig Identities
cos2 x + sin2 x = 1
cos2 x - sin2 x = cos 2x, 2 sin x cos x = sin 2x
cos2 x
=
1
+ cos 2x ,
sin2 x = 1 - cos 2x
2
2
cos(-x) = cos x, sin(-x) = - sin x, tan(-x) = - tan x, cot(-x) = - cot x
cos(x + y) = cos x cos y - sin x sin y, sin(x + y) = sin x cos y + cos x sin y
sin x
1 cos x
tan x = , cot x =
=
cos x
tan x sin x
tan x + tan y tan(x + y) =
1 - tan x tan y
1
1
1
sec x =
, csc x =
=
cos x
sin x sec x
Examples
1.
Solve
the
equation:
sin x
=
1 2
Solution: From the unit circle we get two solutions inside the interval [0, 2):
5
x1 = 6
and x2 =
. 6
Sine is a periodic function with the period 2. So, the actual solutions are made of two infinite sequences of numbers
5 5
5
5
5
, + 2, - 2, + 4, - 4, ... and , + 2, - 2, + 4, - 4, ...
66
6
6
6
66
6
6
6
5
It
briefly can
be
written
as
x1
=
+ 2k, 6
x2
=
+2k, where k is an integer number (positive 6
or negative).
5
Notice that
6
=
-
. 6
Then
x2
=
- 6
+
and
these
two
sequences
of
solutions
can
be
1
combined into a single formula x = (-1)k + k. 6
Answer: x = (-1)k + k. 6
2. Solve the equation: cos x =
3 2
Solution:
From
the
unit
circle
we
get
two
solutions
x1
=
6
and
x2
=- . 6
Cosine
is
a
periodic
function with the period 2. So, the actual solutions are made of two infinite sequences of
numbers
, + 2, - 2, + 4, - 4, ... and - , - + 2, - - 2, - + 4, - - 4, ...
66
6
6
6
66
6
6
6
It
briefly
can
be
written
as
x1
=
6
+ 2k,
x2
=
- 6
+ 2k.
These two sequences of solutions can be combined in a single formula x = ? + 2k.
6
Answer: x = ? + 2k.
6
3.
Solve
the
equation:
sin x
=
-
1 3
Solution: From the unit circle we get two solutions between 0 and 2:
x1 = sin-1
-
1 3
= - sin-1
1 3
and x2 = sin-1
1 3
+ .
The full set of solutions is given by x1 = - sin-1
1 3
+ 2k and x2 = sin-1
1 3
+ + 2k.
These two sequences of solutions can be combined into a single formula
x = (-1)k-1 sin-1
1 3
+ k.
Answer: x = (-1)k-1 sin-1
1 3
+ k.
4.
Solve
the
equation:
cot x - sin x
=
2 sin2
x 2
Solution:
2 sin2
x 2
=
1 - cos x.
Then
cos x
cot x - sin x = 1 - cos x
+ cos x - sin x - 1 = 0
sin x
cos x + cos x sin x - sin x(sin x - 1)
=0
sin x
cos x(1 + sin x) - sin x(sin x + 1) = 0 and sin x = 0
2
(cos x - sin x)(sin x + 1) = 0 and x = n cos x = sin x or sin x = -1 and x = n
x = + k or x = - + 2k and x = n.
4
2
Answer:
x1 = 4 + k,
x2
=
- 2
+ 2k.
5. Prove the identity: cos 2x + 1 = 2 cos2 x Solution: cos 2x = cos2 x - sin2 x, 1 = cos2 x + sin2 x. Then cos 2x + 1 = cos2 x - sin2 x + cos2 x + sin2 x = 2 cos2 x.
6. Prove the identity: tan x + cot x = sec x csc x
Solution:
sin x cos x sin2 x + cos2 x
1
11
tan x + cot x =
+
=
=
=
?
= sec x csc x.
cos x sin x cos x sin x cos x sin x cos x sin x
7. Find the domain of the function: f (x) =
3
(a) when - x
2
2
5
(b) when x
2
2
(c) when 0 x 2
1 2
-
sin
x
Solution:
The
domain
can
be
found
from
the
inequality
1 2
- sin x
0
or
sin x
1 2
.
7
From the unit circle we get the set of intervals of the solution: x - + 2k, + 2k
6
6
7 (a) Take k = 0 to get the answer x - ,
66 5 13 (b) Take k = 1 to get the answer x , 66
(c) Here we take a combination of solutions from parts (a) and (b) to make a solution that lies inside the interval [0, 2].
5
The answer is x 0, , 2
6
6
3
Exersices. Calculators are NOT permitted.
Easy
Prove the identities
1. cos x tan x = sin x
tan x
2.
= sin x
sec x
3. (1 - tan x)2 = sec2 x - 2 tan x
4. (cos x - sin x)2 = 1 - sin 2x
cos x
5.
= tan x + sec x
1 - sin x
Solve the equations. Find all solutions.
2
6. sin x = 2
7. tan x = 3
3
8. sin x = - 2
10. sin 2x cos 2x = 0 11. sin 2x = cos 2x.
1 9. cos 3x = -
2
Medium
Prove the identities
12.
1
1
+
= 2 csc2 x
1 - cos x 1 + cos x
13. (cos x - sin x)2 + (cos x + sin x)2 = 2
14. cot2 x - cos2 x = cot2 x cos2 x
1 + cos x sec x + 1
15.
=
1 - cos x sec x - 1
Solve the equations. Find all solutions.
2
16. sin x cos x = 4
17. 4 cos2 x - 4 cos x + 1 = 0
19. sin4 4x - cos4 4x = 1
1 + tan2 x 20. sin2 x + cos2 x = 2
18. 3 sin x = 2 cos2 x
cot 2x
2
21.
cos2 x - sin2 x
=
. 3
Hard (challenge) problems
Prove the identities
22.
sin 9x + sin 10x + sin 11x + sin 12x
=
4 cos
x 2
cos x sin
21 2
x
23.
cos 2x - cos 3x - cos 4x + cos 5x
=
-4 sin
x 2
sin x cos
7 2
x
4
24. sin2 2x - cos
3
-
2x
sin
2x
-
6
1 =
4
tan 3x 1 - cot2 3x 25. tan2 3x - 1 ? cot 3x = 1
26. 3 - 4 cos 2x + cos 4x = tan4 x 3 + 4 cos 2x + cos 4x
Solve the equations
27. cos 3x - sin x = 3 (cos x - sin 3x)
k
Ans: x1 = 8 + 2 , x2 = 12 + k
28.
1
4 cot x + cot2
x
+
sin2
2x
+
1
=
0
Ans: x = - + k
4
29.
sin 2x
=
cos4
x 2
- sin4
x 2
Ans:
x1
=
2
+ k,
x2
=
(-1)k
6
+ k
30. (1 + cos 4x) sin 2x = cos2 2x
Ans:
x1
=
4
+
k ,
2
x2
=
(-1)k
12
+
k 2
31. tan x - tan 2x = sin x Ans: x = k
32.
3 sin2 x + 3 cos2 x =
34
k Ans: x = +
42
In all answers k is an integer number.
Find the domain of the functions
33. f (x) =
cos
x
-
1 2
when
- x
2
2
34. f (x) =
3 2
-
cos
x
when
0 x 2
3
35. f (x) = 2 - tan x when x 0, ,
2
22
5
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.