TAREA DE TRIGONOMETRÍATEMAS: IDENTIDADES-ECUACIONES TRIGONOMÉTRICAS-LEY DE SENOS Y

COSENOS

1. The following diagram shows two semi-circles. The larger one has centre O and radius 4 cm. The smaller one has centre P, radius 3 cm, and passes through O. The line (OP) meets the larger semi-circle at S. The semi-circles intersect at Q.

(a) (i) Explain why OPQ is an isosceles triangle.

(ii) Use the cosine rule to show that cos QPO = 9

1

.

(iii) Hence show that sin QPO = 9

80

.(iv) Find the area of the triangle OPQ.

(b) Consider the smaller semi-circle, with centre P.

(i) Write down the size of Q.PO

(ii) Calculate the area of the sector OPQ.

(c) Consider the larger semi-circle, with centre O. Calculate the area of the sector QOS.

(d) Hence calculate the area of the shaded region.

2. The diagram below shows triangle PQR. The length of [PQ] is 7 cm, the length of [PR] is 10

cm, and RQP is 75.

(a) Find R.QP

(b) Find the area of triangle PQR.

3. In triangle PQR, PQ is 10 cm, QR is 8 cm and angle PQR is acute. The area of the triangle is 20

cm2. Find the size of angle R.QP

Una comunidad en busca de la verdad

4. Town A is 48 km from town B and 32 km from town C as shown in the diagram.

AB

C

3 2 k m

4 8 k m

Given that town B is 56 km from town C, find the size of angle BAC to the nearest degree.

5. The diagram below shows a quadrilateral ABCD. AB = 4, AD = 8, CD =12, B C D = 25, DAB  =.

(a) Use the cosine rule to show that BD = cos454 .Let = 40.

(b) (i) Find the value of sin DBC .

(ii) Find the two possible values for the size of DBC .

(iii) Given that DBC is an acute angle, find the perimeter of ABCD. (c) Find the area of triangle ABD.

6. The following diagram shows a triangle ABC, where BC = 5 cm, B = 60°, C = 40°.

6 0 ° 4 0 °

A

B C5 cm

(a) Calculate AB.(b) Find the area of the triangle.

7. In a triangle ABC, AB = 4 cm, AC = 3 cm and the area of the triangle is 4.5 cm2.

Find the two possible values of the angle CAB .

Una comunidad en busca de la verdad 2

8. In triangle ABC, AC = 5, BC = 7, A = 48°, as shown in the diagram.

A B

C

5 7

4 8 °

d iag ra m n o t to sca le

Find ,B giving your answer correct to the nearest degree. 9. Two boats A and B start moving from the same point P. Boat A moves in a straight line at

20 km h–1 and boat B moves in a straight line at 32 km h–1. The angle between their paths is 70°.Find the distance between the boats after 2.5 hours.

10. Consider the equation 3 cos 2x + sin x = 1

(a) Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r , and p , q , r .

(b) Factorize f (x).(c) Write down the number of solutions of f (x) = 0, for 0 x 2.

11. The diagrams below show two triangles both satisfying the conditions

AB = 20 cm, AC = 17 cm, CBA = 50°.Diagrams not

to scale

A

B C

A

B C

Trian g le 1 Trian g le 2

(a) Calculate the size of BCA in Triangle 2.(b) Calculate the area of Triangle 1.

12. A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the largest angle.

13. The following diagram shows a triangle with sides 5 cm, 7 cm, 8 cm.

5 7

8 Diagram not to scale

Una comunidad en busca de la verdad 3

Find(a) the size of the smallest angle, in degrees;(b) the area of the triangle.

Una comunidad en busca de la verdad 4

14. The function f is defined by f : x 30 sin 3x cos 3x, 0 x 3

π

.(a) Write down an expression for f (x) in the form a sin 6x, where a is an integer.(b) Solve f (x) = 0, giving your answers in terms of .

15. Consider y = sin

9x

.(a) The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where 0  x  π.

(b) Solve the equation sin

9x

= – 21

, for 0 x 2.

16. Consider the trigonometric equation 2 sin2 x = 1 + cos x.

(a) Write this equation in the form f (x) = 0, where f (x) = a cos2 x + b cos x + c,and a, b, c .

(b) Factorize f (x).(c) Solve f (x) = 0 for 0° x 360°.

17. Solve the equation 2 cos2 x = sin 2x for 0 x π, giving your answers in terms of π.

18. Solve the equation 3 sin2 x = cos2 x, for 0° x 180°.

19. (a) Factorize the expression 3 sin2 x – 11 sin x + 6.

(b) Consider the equation 3 sin2 x – 11 sin x + 6 = 0.(i) Find the two values of sin x which satisfy this equation,(ii) Solve the equation, for 0° x 180°.

20. Let f (x) = sin 2x and g (x) = sin (0.5x).

(a) Write down(i) the minimum value of the function f ;(ii) the period of the function g.

(b) Consider the equation f (x) = g (x).

Find the number of solutions to this equation, for 0 x 2

π3

.

21. Solve the equation 3 cos x = 5 sin x, for x in the interval 0° x 360°, giving your answers to the nearest degree.

22. (a) Express 2 cos2 x + sin x in terms of sin x only.

(b) Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0 x , giving your answers exactly.

23. (a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. (b) Hence or otherwise, solve the equation

3 sin2 x + 4 cos x – 4 = 0, 0 x 90.

Una comunidad en busca de la verdad 5

24. In triangle ABC, AB = 9 cm, AC =12 cm, and B is twice the size of C .

Find the cosine of C .

25. Consider triangle ABC with CABˆ = 37.8, AB = 8.75 and BC = 6.

Find AC.

26. Triangle ABC has C = 42, BC =1.74 cm, and area 1.19 cm2.

(a) Find AC.

(b) Find AB.

27. The following diagram shows ABC, where BC =105 m, BCAˆ = 40, CBAˆ

= 60.

Find AB.

28. In triangle ABC, CBA = 31, AC = 3 cm and BC = 5 cm. Calculate the possible lengths of the side [AB].

29. Triangle ABC has AB = 8 cm, BC = 6 cm and CAB = 20°. Find the smallest possible area of ABC.

30. In the triangle ABC, A = 30°, BC = 3 and AB = 5. Find the two possible values of B .

31. Find all the values of θ in the interval [0, ] which satisfy the equation: cos 2 = sin2 .

32. Solve tan2 2 =1, in the interval .θ

2

π

2

π

33. The angle θ satisfies the equation 2 tan2 θ – 5 sec θ – 10 = 0, where θ is in the second quadrant. Find the exact value of sec θ.

Una comunidad en busca de la verdad 6

34. The angle satisfies the equation tan + cot = 3, where is in degrees. Find all the possible values of lying in the interval [0°, 90°].

35. Solve 2 sin x = tan x, where 2–

x .

2

36. Let ,A ,B C be the angles of a triangle. Show that tan A + tan B + tan C = A tan B tan .C

37. In the diagram below, AD is perpendicular to BC.

CD = 4, BD = 2 and AD = 3. DAC ˆ = and DABˆ = .

Find the exact value of cos ( − ).

38. Prove that

4cos–12cos

2cos–14sin

= tan , for 0 < < 2

π

, and 4

π

.

39. Given that a sin 4x + b sin 2x = 0, for 0 < x < 2

, find an expression for cos2 x in terms of a and b.

40. In the obtuse-angled triangle ABC, AC = 10.9 cm, BC = 8.71 cm and CABˆ = 50.

A

B

C5 0 °

1 0 .9 cm

8 .7 1 cm

N o t tosca le

Find the area of triangle ABC.

Una comunidad en busca de la verdad 7