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2
Introductory Problems
1. Let x be a real number such that sec x tan x = 2. Evaluate sec x + tan x.
2. Let 0 < < 45. Arrange
t1 = (tan )tan , t2 = (tan )cot ,t3 = (cot )tan , t4 = (cot )cot ,
in decreasing order.
3. Compute
(a) sin 12
, cos 12
, and tan 12
;
(b) cos4 24
sin4 24
;
(c) cos 36 cos 72; and(d) sin 10 sin 50 sin 70.
4. Simplify the expressionsin4 x + 4 cos2 x
cos4 x + 4 sin2 x.
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64 103 Trigonometry Problems
5. Prove that
1 cot 23 =2
1 cot 22 .
6. Find all x in the interval
0, 2
such that
3 1
sin x+
3 + 1
cos x= 4
2.
7. RegionR contains all the points (x,y) such that x2 + y2 100 and sin(x +y) 0. Find the area of region R.
8. In triangle ABC, show that
sinA
2 a
b + c .
9. Let I denote the interval
4,
4
. Determine the function f defined on the
interval [1, 1] such that f (sin 2x) = sin x + cos x and simplify f (tan2 x)for x in the interval I.
10. Let
fk (x)
=
1
k
(sink x
+cosk x)
for k = 1, 2, . . . . Prove that
f4(x) f6(x) =1
12
for all real numbers x.
11. A circle of radius 1 is randomly placed in a 15-by-36 rectangle ABCD so that
the circle lies completely within the rectangle. Given that the probability thatthe circle will not touch diagonal AC is m
n, where m and n are relatively prime
positive integers, find m + n.
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2. Introductory Problems 65
12. In triangle ABC,
3sin A + 4cos B = 6 and 4 sin B + 3cos A = 1.
Find the measure of angle C.
13. Prove that
tan 3a tan 2a tan a = tan 3a tan 2a tan afor all a = k
2, where k is in Z.
14. Let a , b , c , d be numbers in the interval [0, ] such that
sin a + 7sin b = 4(sin c + 2sin d),cos a + 7cos b = 4(cos c + 2cos d).
Prove that 2 cos(a d) = 7cos(b c).
15. Expresssin(x y) + sin(y z) + sin(z x)
as a monomial.
16. Prove that
(4cos2 9 3)(4cos2 27 3) = tan 9.
17. Prove that 1 + a
sin x
1 + b
cos x
1 +
2ab2
for all real numbers a , b , x with a, b 0 and 0 < x < 2
.
18. In triangle ABC, sin A + sin B + sin C 1. Prove that
min
{A
+B, B
+C, C
+A
}< 30.
19. Let ABC be a triangle. Prove that
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66 103 Trigonometry Problems
(a)
tan A2
tan B2
+ tan B2
tan C2
+ tan C2
tan A2
= 1;
(b)
tanA
2tan
B
2tan
C
2
3
9.
20. Let ABC be an acute-angled triangle. Prove that
(a) tan A + tan B + tan C = tan A tan B tan C;(b) tan A tan B tan C 3
3.
21. Let ABC be a triangle. Prove that
cot A cot B + cot B cot C + cot C cot A = 1.
Conversely, prove that ifx , y , z are real numbers with xy + yz + zx = 1, thenthere exists a triangle ABC such that cot A = x, cot B = y, and cot C = z.
22. Let ABC be a triangle. Prove that
sin2A
2+ sin2 B
2+ sin2 C
2+ 2sin A
2sin
B
2sin
C
2= 1.
Conversely, prove that ifx , y , z are positive real numbers such that
x2
+y2
+z2
+2xyz
=1,
then there is a triangle ABC such that x = sin A2
, y = sin B2
, and z = sin C2
.
23. Let ABC be a triangle. Prove that
(a) sinA
2sin
B
2sin
C
2 1
8;
(b) sin2A
2+ sin2 B
2+ sin2 C
2 3
4;
(c) cos2 A2
+ cos2 B2
+ cos2 C2
94
;
(d) cosA
2cos
B
2cos
C
2 3
3
8;
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2. Introductory Problems 67
(e) cscA
2 +csc
A
2 +csc
A
2 6.
24. In triangle ABC, show that
(a) sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C;(b) cos 2A + cos2B + cos 2C = 1 4cos A cos B cos C;(c) sin2 A + sin2 B + sin2 C = 2 + 2cos A cos B cos C;(d) cos2 A + cos2 B + cos2 C + 2 cos A cos B cos C = 1.
Conversely, ifx , y , z are positive real numbers such that
x2 + y2 + z2 + 2xyz = 1,
show that there is an acute triangle ABC such that x = cos A, y = cos B,C = cos C.
25. In triangle ABC, show that
(a) 4R = abc[ABC] ;
(b) 2R2 sin A sin B sin C = [ABC];(c) 2R sin A sin B sin C = r(sin A + sin B + sin C);
(d) r = 4R sin A2
sinB
2sin
C
2;
(e) a cos A + b cos B + c cos C = abc2R2
.
26. Let s be the semiperimeter of triangle ABC. Prove that
(a) s = 4R cos A2
cosB
2cos
C
2;
(b) s 3
3
2R.
27. In triangle ABC, show that
(a) cos A + cos B + cos C = 1 + 4sin A2
sinB
2sin
C
2;
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68 103 Trigonometry Problems
(b) cos A
+cos B
+cos C
3
2
.
28. Let ABC be a triangle. Prove that
(a) cos A cos B cos C 18
;
(b) sin A sin B sin C 3
3
8;
(c) sin A + sin B + sin C 3
3
2 .
(d) cos2 A + cos2 B + cos2 C 34
;
(e) sin2 A + sin2 B + sin2 C 94
;
(f) cos 2A + cos2B + cos 2C 32
;
(g) sin 2A + sin 2B + sin 2C 3
3
2 .
29. Prove thattan 3x
tan x= tan
3
x
tan
3+ x
for all x = k6
, where k is in Z.
30. Given that
(1 + tan 1)(1 + tan 2) (1 + tan 45) = 2n,
find n.
31. Let A = (0, 0) and B = (b, 2) be points in the coordinate plane. Let ABCDEFbe a convex equilateral hexagon such that F AB = 120, AB DE,BC EF, and CD F A, and the y coordinates of its vertices are distinct elementsof the set {0, 2, 4, 6, 8, 10}. The area of the hexagon can be written in the formm
n, where m and n are positive integers and n is not divisible by the square
of any prime. Find m + n.
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2. Introductory Problems 69
32. Show that one can use a composition of trigonometry buttons, such as sin, cos,
tan, sin1
, cos1
, and tan1
, to replace the broken reciprocal button on acalculator.
33. In triangle ABC, A B = 120 and R = 8r. Find cos C.
34. Prove that in a triangle ABC,
a b
a + b =tan
A B
2
tanC
2
.
35. In triangle ABC, ab
= 2 +
3 and C = 60. Find the measure of angles Aand B.
36. Let a , b , c be real numbers, all different from 1 and 1, such that a + b + c =abc. Prove that
a1 a2 +
b1 b2 +
c1 c2 =
4abc(1 a2)(1 b2)(1 c2) .
37. Prove that a triangle ABC is isosceles if and only if
a cos B + b cos C + c cos A = a + b + c2
.
38. Evaluate
cos a cos 2a cos3a cos 999a,where a = 2
1999.
39. Determine the minimum value of
sec4
tan2 +sec4
tan2
over all , = k2
, where k is in Z.
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70 103 Trigonometry Problems
40. Find all pairs (x,y) of real numbers with 0 < x < 2
such that
(sin x)2y
(cos x)y2/2
+ (cos x)2y
(sin x)y2/2
= sin 2x.
41. Prove that cos 1 is an irrational number.
42. Find the maximum value of
S = (1 x1)(1 y1) + (1 x2)(1 y2)
ifx21 + x22 = y21 + y22 = c2.
43. Prove thatsin3 a
sin b+ cos
3 a
cos b sec(a b)
for all 0 < a, b Tn(x) for real numbers x with x > 1;(e) Determine all the roots ofTn(x);
(f) Determine all the roots ofPn(x) = Tn(x) 1.
50. Let ABC be a triangle with BAC = 40 and ABC = 60. Let D and E bethe points lying on the sides AC and AB, respectively, such that CBD = 40and BCE = 70. Segments BD and CE meet at F. Show that AF BC.
51. Let S be an interior point of triangle ABC. Show that at least one of SAB, SBC, and SCA is less than or equal to 30.
52. Let a=
7
.
(a) Show that sin2 3a sin2 a = sin 2a sin 3a;(b) Show that csc a = csc2a + csc4a;(c) Evaluate cos a cos2a + cos 3a;(d) Prove that cos a is a root of the equation 8x3 + 4x2 4x 1 = 0;(e) Prove that cos a is irrational;
(f) Evaluate tan a tan 2a tan 3a;
(g) Evaluate tan2 a+
tan2 2a+
tan2 3a;
(h) Evaluate tan2 a tan2 2a + tan2 2a tan2 3a + tan2 3a tan2 a.(i) Evaluate cot2 a + cot2 2a + cot2 3a.