Trigonometry Problems - Amir Hossein Parvardi.pdf

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Trigonometry Problems
Amir Hossein Parvardi
February 16, 2011
1. Prove that:
P
cos 2
13
+ cos 6
13 + cos 8
13−1
4
=
13
cos 4 19 + cos 6 19 + cos 10
2. Prove that 2
is a root of the equation:
19
r
q
P
4 +
4 +
4−x = x
3. Prove that
t
s
r
1
2 + 1
1
2 + 1
1
2 + 1
2 cos 8 = cos
2
2
4. Prove that
8
3
8
5
8
7
8
= 3
2
sin 4
+ sin 4
+ sin 4
+ sin 4
5. Prove that
x
2
x
4
x
8
sin 2x
16 sin
cos x·cos
·cos
·cos
=
8
6. Prove that
3
4
64·sin 10 ◦ ·sin 20 ◦ ·sin 30 ◦ ·sin 40 ◦ ·sin 50 ◦ ·sin 60 ◦ ·sin 70 ◦ ·sin 80 ◦ ·sin 90 ◦ =
1
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7. Find x if
sin x = tan 12 ◦ ·tan 48 ◦ ·tan 54 ◦ ·tan 72 ◦ ·
8. Solve the following equations in R:
•sin 9x + sin 5x + 2 sin 2 x = 1
•cos 5x·cos 3x−sin 3x·sin x = cos 2x
4
•cos 5x + cos 3x + sin 5x + sin 3x = 2·cos
−4x
•sin x + cos x−sin x·cos x =−1
•sin 2x− P
3 cos 2x = 2
9. Prove following equations:
2 7
4 7
6 7
7
3 7
5 7
•sin
+ sin
−sin
= 4 sin
·sin
·sin
13
3 13
5 13
7 13
9 13
11
13
1
2
•cos
+ cos
+ cos
+ cos
+ cos
+ cos
=
(2K−1)
2K+1
2K+1
3
2K+1
= 2
•8k2
N
: cos
+ cos
+···+ cos
= 1
4
7
2 7
3 7
4
•sin
+ sin
+ sin
·cot
10. Show that
cos
n + cos 2
+···+ cos n
n
=−1.
n
SIN 2 NA
11.Show that cos a + cos 3a + cos 5a +···+ cos(2n−1)a =
2 SIN A .
SIN 2 NA
SIN A
12. Show that sin a + sin 3a + sin 5a +···+ sin(2n−1)a =
.
13. Calculate
(tan 1 ◦ ) 2 + (tan 2 ◦ ) 2 + (tan 3 ◦ ) 2 + . . . + (tan 89 ◦ ) 2 .
14. Prove that cot 2 7 + cot 2 2 7
+ cot 2 3 7
= 5.
P
15. Show that tan 7 tan 2 7
tan 3 7
=
7.
2
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, cos
2 7
4 7
6 7
16. cos
are the roots of an equation of the form
ax 3 + bx 2 + cx + d = 0 where a, b, c, d are integers. Determine a, b, c and d.
and cos
*17. Find the value of the sum
r
r
r
cos 2
7
cos 4
7
cos 6
7
3
+
3
+
3
.
18. Solve the equation
2 sin 4 x(sin 2x−3)−2 sin 2 x(sin 2x−3)−1 = 0.
19. Express the sum of the following series in terms of sin x and cos x.
X
N
x + k
(2k + 1) sin 2
2
K=0
20. Find the smallest positive integer N for which
1
sin 45 ◦ ·sin 46 ◦ +
1
sin 47 ◦ ·sin 48 ◦ +···+
1
sin 133 ◦ ·sin 134 ◦ =
1
sin N ◦ .
21. Find the value of
sin 40 ◦ + sin 80 ◦
sin 110 ◦ .
22. Evaluate the sum
S = tan 1 ◦ ·tan 2 ◦ + tan 2 ◦ ·tan 3 ◦ + tan 3 ◦ ·tan 4 ◦ +···+ tan 2004 ◦ ·tan 2005 ◦ .
23. Solve the equation :
P
P
P
P
3 sin x(cos x−sin x) + (2−
6) cos x + 2 sin x +
3−2
2 = 0.
1
SIN 7
24. Let f (x) =
. Prove that f (3) + f (2) = f (1).
25. Suppose that real numbers x, y, z satisfy
cos x + cos y + cos z
cos (x + y + z)
sin x + sin y + sin z
sin (x + y + z)
=
= p
3
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Prove that
cos (x + y) + cos (y + z) + cos (x + z) = p.
26. Solve for , 0 2 :
sin 5 + cos 5 = 1.
27. For x, y2[0, 3 ] prove that cos x + cos y 1 + cos xy.
28. Prove that among any four distinct numbers from the interval (0, 2 ) there
are two, say x, y, such that:
8 cos x cos y cos(x−y) + 1 > 4(cos 2 x + cos 2 y).
29. Let B = 7 . Prove that
tan B·tan 2B + tan 2B·tan 4B + tan 4B·tan B =−7.
30. a) Calculate
1
cos 6 13
−4 cos 4
13
−4 cos 5
13
=?
b) Prove that
13 + 4 sin 4
= tan 3
13 + 4 sin 3
tan
13
13
c) Prove that
tan 2
13
+ 4 sin 6
13
= tan 5
13 + 4 sin 2
13
cos 2 + cos 2
31. Prove that if , are angles of a triangle and
(1 + tan ·tan ) =
2, then + = 90 ◦ .
32. Let a, b, c, d2[− 2 , 2 ] be real numbers such that sin a+sin b+sin c+sin d = 1
and cos 2a + cos 2b + cos 2c + cos 2d 1 3 . Prove that a, b, c, d2[0, 6 ]
33. Find all integers m, n for which we have sin M x + cos N x = 1, for all x.
34. Prove that tan 55 ◦ ·tan 65 ◦ ·tan 75 ◦ = tan 85 ◦ .
4 COS 12 ◦ +4 COS 36 ◦ +1
√
3
= tan 78 ◦ .
35. Prove that
4
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36. Prove that
t
t
s
r
q
P
cos 4
19 + cos 6
+ cos 10
19
4 +
4 +
4−
4 +
4 +
4−···= 2
.
19
The signs: + +−+ +−+ +−+ +−···
37. For reals x, y Prove that cos x + cos y + sin x sin y 2.
38. Solve the equation in real numbers
r
q
P
7 + 2
7−2
7−2x = x.
39. Let A, B, C be three angles of triangle ABC. Prove that
(1−cos A)(1−cos B)(1−cos C) cos A cos B cos C.
40. Solve the equation
sin 3 (x)−cos 3 (x) = sin 2 (x) .
P
N
K=1 sin 2 k for n > 1
42. Prove the following without using induction:
41. Find S N =
cos N +1
2
x·sin 2 x
sin 2
cos x + cos 2x +···+ cos nx =
.
43. Evaluate:
sin + 1
2
1
2 2
1
2 3
·sin 2 +
·sin 3 +
·sin 4 +···
44. Compute
N−1
X
k
n
csc 2
.
K=1
45. Prove that
h
i
+ (N−1)
N
+ N
+ 2 N
n + N 2
• tan +tan
+tan
+···+tan
=−n cot
.
5
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