HL Week 2 Revision – Trigonometry Questions 1a. [3 marks] Consider Express w 2 and w 3 in modulus-argument form. 1b. [2 marks] Sketch on an Argand diagram the points represented by w 0 , w 1 , w 2 and w 3 . 1c. [3 marks] These four points form the vertices of a quadrilateral, Q. Show that the area of the quadrilateral Q is . 1d. [6 marks] Let . The points represented on an Argand diagram by form the vertices of a polygon . Show that the area of the polygon can be expressed in the form , where . 2. [5 marks] Let . Find, in terms of b, the solutions of . 3. [5 marks] Let where . Express in terms of sin and cos . 1
Transcript
HL Week 2 Revision – Trigonometry Questions
1a. [3 marks]
Consider
Express w2 and w3 in modulus-argument form.
1b. [2 marks]
Sketch on an Argand diagram the points represented by w0 , w1 , w2 and w3.
1c. [3 marks]
These four points form the vertices of a quadrilateral, Q.
Show that the area of the quadrilateral Q is .
1d. [6 marks]
Let . The points represented on an Argand diagram
by form the vertices of a polygon .
Show that the area of the polygon can be expressed in the form , where .
2. [5 marks]
Let .
Find, in terms of b, the solutions of .
3. [5 marks]
Let where .
Express in terms of sin and cos .
4a. [3 marks]
Consider the following diagram.
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The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The
circular arc AB has centre, M, the midpoint of [CP].
Find AM.
4b. [2 marks]
Find in radians.
4c. [3 marks]
Find the area of the shaded region.
5a. [2 marks]
Consider the function .
Determine an expression for in terms of .
5b. [4 marks]
Sketch a graph of for .
5c. [2 marks]
Find the -coordinate(s) of the point(s) of inflexion of the graph of , labelling these clearly on
the graph of .
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5d. [2 marks]
Let .
Express in terms of .
5e. [3 marks]
Express in terms of .
5f. [2 marks]
Hence show that can be expressed as .
5g. [3 marks]
Solve the equation , giving your answers in the form where .
6. [4 marks]
This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius
and four equal sectors of a smaller circle of radius .
The angle 20°.
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Find the area of the pendant.
7. [6 marks]
Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.
“Seaview” is at an angle of depression of 25°.
“Nauti Buoy” is at an angle of depression of 35°.
The following three dimensional diagram shows Barry and the two yachts at S and N.
X lies at the foot of the cliff and angle 70°.
Find, to 3 significant figures, the distance between the two yachts.
8. [5 marks]
Solve the equation .
9a. [5 marks]
Solve .
9b. [3 marks]
Show that .
9c. [9 marks]
4
Let .
Find the modulus and argument of in terms of . Express each answer in its simplest form.
9d. [5 marks]
Hence find the cube roots of in modulus-argument form.
10a. [2 marks]
In triangle and .
Use the cosine rule to show that .
10b. [3 marks]
Consider the possible triangles with .
Calculate the two corresponding values of PQ.
10c. [3 marks]
Hence, find the area of the smaller triangle.
10d. [7 marks]
Consider the case where , the length of QR is not fixed at 8 cm.
Determine the range of values of for which it is possible to form two triangles.
11a. [3 marks]
A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with
radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section.
The centre of the circle is O and the angle KOL is radians.
5
Find an expression for the volume of water in the trough in terms of .
11b. [4 marks]
The volume of water is increasing at a constant rate of .
Calculate when .
12a. [2 marks]
Find the set of values of that satisfy the inequality .
12b. [4 marks]
The triangle ABC is shown in the following diagram. Given that , find the range of possible
values for AB.
13a. [2 marks]
Find the value of .
13b. [2 marks]
Show that where .
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13c. [9 marks]
Use the principle of mathematical induction to prove that
where .
13d. [6 marks]
Hence or otherwise solve the equation in the interval .
14a. [3 marks]
Consider the function defined by where .
Sketch the graph of indicating clearly any intercepts with the axes and the coordinates of any local
maximum or minimum points.
14b. [2 marks]
State the range of .
14c. [4 marks]
Solve the inequality .
15a. [5 marks]
In a triangle and .
Use the cosine rule to find the two possible values for AC.
15b. [3 marks]
Find the difference between the areas of the two possible triangles ABC.
16a. [3 marks]
The diagram shows two circles with centres at the points A and B and radii and , respectively. The
point B lies on the circle with centre A. The circles intersect at the points C and D.
7
Let be the measure of the angle CAD and be the measure of the angle CBD in radians.
Find an expression for the shaded area in terms of , and .
16b. [2 marks]
Show that .
16c. [3 marks]
Hence find the value of given that the shaded area is equal to 4.
17a. [2 marks]
The following diagram shows the curve , where , , and are all positive
constants. The curve has a maximum point at and a minimum point at .
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Write down the value of and the value of .
17b. [2 marks]
Find the value of .
17c. [2 marks]
Find the smallest possible value of , given .
18a. [1 mark]
Expand and simplify .
18b. [3 marks]
By writing as find the value of .
18c. [4 marks]
The following diagram shows the triangle ABC where and .
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Find BC in the form where .
19a. [1 mark]
Show that for .
19b. [4 marks]
Hence find .
20a. [3 marks]
Consider the equation . Given that and
verify that is a solution to the equation;
20b. [5 marks]
hence find the other solution to the equation for .
21. [6 marks]
The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The points
A and C are 3 m apart. A goat is tied by a 5 m length of rope at point A on the outside edge of the
enclosure.
Given that the corner of the enclosure at C forms an angle of radians and the area of field that can be
reached by the goat is , find the value of .
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22. [5 marks]
ABCD is a quadrilateral where and .
Find , giving your answer correct to the nearest degree.
23a. [4 marks]
Points A , B and T lie on a line on an indoor soccer field. The goal, [AB] , is 2 metres wide. A player
situated at point P kicks a ball at the goal. [PT] is perpendicular to (AB) and is 6 metres from a parallel
line through the centre of [AB] . Let PT be metros and let measured in degrees. Assume
that the ball travels along the floor.
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Find the value of when .
23b. [4 marks]
Show that .
23c. [11 marks]
The maximum for gives the maximum for .
(i) Find .
(ii) Hence or otherwise find the value of such that .
(iii) Find and hence show that the value of never exceeds 10°.
23d. [3 marks]
Find the set of values of for which .
24. [4 marks]
12
The following diagram shows a sector of a circle where radians and the length of the
.
Given that the area of the sector is , find the length of the arc .
25a. [1 mark]
Show that .
25b. [7 marks]
Consider where is a constant. Prove by mathematical induction that
where and represents the derivative of .
26. [7 marks]
Solve the equation for .
27a. [4 marks]
A function is defined by , where . The following
diagram represents the graph of .
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Find the value of
(i) ;
(ii) ;
(iii) .
27b. [2 marks]
Solve for .
28. [6 marks]
Triangle has area . The sides and have lengths cm and cm respectively.
Find the two possible lengths of the side .
29a. [3 marks]
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The logo, for a company that makes chocolate, is a sector of a circle of radius cm, shown as shaded in
the diagram. The area of the logo is .
Find, in radians, the value of the angle , as indicated on the diagram.
29b. [2 marks]
Find the total length of the perimeter of the logo.
30a. [2 marks]
Consider the functions and .
Find an expression for , stating its domain.
30b. [2 marks]
Hence show that .
30c. [6 marks]
Let , find an exact value for at the point on the graph of where ,
expressing your answer in the form .
30d. [6 marks]
15
Show that the area bounded by the graph of , the -axis and the lines and is
.
31. [6 marks]
Find all solutions to the equation where .
32a. [4 marks]
In triangle , and .
Show that length .
32b. [4 marks]
Given that has a minimum value, determine the value of for which this occurs.
33a. [6 marks]
(i) Use the binomial theorem to expand .
(ii) Hence use De Moivre’s theorem to prove
(iii) State a similar expression for in terms of and .
33b. [4 marks]
Let , where is measured in degrees, be the solution of which has
the smallest positive argument.
Find the value of and the value of .
33c. [4 marks]
Using (a) (ii) and your answer from (b) show that .
33d. [5 marks]
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Hence express in the form where .
34. [6 marks]
A triangle has , and . Find the area of the triangle given that it
