fulltextproblemas introductorios trigonometria

8/6/2019 fulltextproblemas introductorios trigonometria

1/9

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.


2/9

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.


3/9

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


4/9


(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.


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.


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

.


(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;


5/9


(e) cscA

2 +csc

A

2 +csc

A

2 6.


(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.


(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.


(a) cos A + cos B + cos C = 1 + 4sin A2

sinB

2sin

C

2;


6/9


(b) cos A

+cos B

+cos C

3

2

.


(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.


7/9


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.


8/9


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.

fulltextproblemas introductorios trigonometria

Documents