Razones Trigonometricas

TAREA DE TRIGONOMETRÍA
TEMAS: 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 = .
(iii) Hence show that sin = .
(iv) Find the area of the triangle OPQ.

(b) Consider the smaller semi-circle, with centre P.
(i) Write down the size of
(ii) Calculate the area of thesector 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 is 75.

(a) Find(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









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


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

5. The diagram below shows a quadrilateral ABCD. AB = 4, AD = 8, CD =12, BD = 25,  =.

(a) Use the cosine rule to show that BD = .Let  = 40.
(b) (i) Find the value of sin .
(ii) Find the two possible values for the size of .
(iii) Given that 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, = 60°, = 40°.(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 .







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

Find 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 theboats 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. Thediagrams below show two triangles both satisfying the conditions
AB = 20 cm, AC = 17 cm, = 50°.
Diagrams not
to scale


(a) Calculate the size of 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...