Title HL Maths Trig Assignment Trigonometric Functions Elementary Mathematics Elementary Geometry 126.8 KB 7
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Module 3 Assignment – Trigonometry

1. (a) Sketch the graph of f(x) = sin 3x + sin 6x, 0 < x < 2.

(b) Write down the exact period of the function f.

Working:

(a) …………………………………………..

(b) ..................................................................

(Total 6 marks)

2. (a) Sketch the graph of the function

xxxC 2cos
2

1
cos)( 

for –2  x  2

(5)

(b) Prove that the function C(x) is periodic and state its period.

(3)

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(c) For what values of x, –2  x  2, is C(x) a maximum?

(2)
(d) Let x = x0 be the smallest positive value of x for which C(x) = 0. Find an approximate

value of x0 which is correct to two significant figures.

(2)

(e) (i) Prove that C(x) = C(–x) for all x.

(2)

(ii) Let x = x1 be that value of x,  < x < 2, for which C(x) = 0. Find the value of x1 in
terms of x0.

(2)
(Total 16 marks)

3. Solve 2 sin x = tan x, where
2

 x  .
2

Working:

…………………………………………..

(Total 3 marks)

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

Working:

………………………………………………..

(Total 6 marks)

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

ABC.

Working:

…………………………………………..

(Total 6 marks)

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6. Find all the values of in the interval [0, ] which satisfy the equation

cos2 = sin2.

Working:

…....…………………………………………..

……..................................................................

(Total 6 marks)

7. In the triangle ABC, Â = 30°, BC = 3 and AB = 5. Find the two possible values of B̂ .

Working:

…....…………………………………………..

……..................................................................

(Total 6 marks)

8. Prove that
 
 4cos–12cos

2cos–14sin
= tan , for 0 <  <

2

π
, and  

4

π
.

(Total 5 marks)

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9. The diagram below shows a circle centre O and radius OA = 5 cm. The angle AOB = 135°.

O

A

B

Find the area of the shaded region.

Working:

…………………………………………........
(Total 6 marks)

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

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(Total 6 marks)

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11. A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the
angle between these two sides is 60°.

(a) Calculate the length of the third side of the field.

(3)

(b) Find the area of the field in the form p 3 , where p is an integer.

(3)

Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into
two parts by constructing a straight fence [AD] of length x metres.

(c) (i) Show that the area of the smaller part is given by
4

65x
and find an expression for

the area of the larger part.

(ii) Hence, find the value of x in the form q 3 , where q is an integer.

(8)

(d) Prove that
8

5

DC

BD
 .

(6)

(Total 20 marks)