Download - 1.TAREA DE TRIGONOMETRÍA
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 .
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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
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Find(a) the size of the smallest angle, in degrees;(b) the area of the triangle.
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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.
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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.
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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.
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