Download NTM3B_supp_E09_OEQ

9.30
New Trend Mathematics S3B — Junior Form Supplementary Exercises
OPEN-ENDED QUESTION ZONE
Write down two sets of x and y which satisfy the equation sin x  cos y .
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
S UGGESTED SOLUTION
[ Analysis: Since the form of sin x  cos y is close to that of the identity sin x  cos(90  x) , we may make use
of the identity to tackle this question. ]
From the identity sin x  cos(90  x),
we have sin 31  cos(90  31)
 cos 59;
sin 63  cos(90  63)
 cos 27
 x  31, y  59 and x  63, y  27 are two sets of x and y.
S OLUTION
OF A STUDENT
30 
45
s in
1
2
1
cos
3
2
1
ta n
1
3

/
2
2
3
3
60
/
2
2
3
2
/
2
2
1
2
1
3
s in x = c o s y
s in 3 0  = c os 60 

x = 30
y = 60

 
s in x = c o s y
s in 6 0  = c os 30 
x = 60
y = 30
Comments
This student can give correct answers and explain clearly.
Chapter 9 Applications of Trigonometry
9.31
The figure shows a straight river. If it is impossible to measure the width of the river directly,
describe a possible method to find its width.
5 marks: 3 marks for correct answer; 2 marks for clear explanation.
S UGGESTED SOLUTION
[ Analysis: We may try to make use of a right-angled triangle in trigonometry. Take some measurements on one
river bank such that one of the sides and one interior angle other than right angle of the right-angled triangle
are known, then the remaining sides can be calculated accordingly. ]
We may make use of a right-angled triangle to calculate the width of the river. The method is as
follows.
1. Select location A on a bank of the river first, then select location B on the other bank of the
river directly opposite to location A ( i.e. AB is the width of the river ).
A
B
2. Mark location C next to location A on the same river bank.
A
C
B
3. Measure the length of AC and ACB,
AB
then tan ACB 
AC
AB  AC tan ACB
S OLUTION
OF A STUDENT
2 cm, measured by a ruler directly.
 
(Translated from a student's solution in Chinese)
9.32
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Comments
This student misinterprets the width required in the question as the width of the river in the given figure, but
not the width of an actual river.
Exercise of Open-ended Questions 9
Level 1
1. Write down two expressions involving the trigonometric ratios of special angles of which
the value of each expression is 3.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
2. Write down two sets of x and y satisfying the equation sin 2 x  sin 2 2 y  1 .
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
3. Sam walked up a slope with the inclination of 5, and then walked up a slope with the
inclination of 10. If the vertical distance between the final position and the starting point
was 100 m, how far did Sam travel?
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
Level 2
4. Describe a possible method to measure the height of the building you live.
5 marks: 3 marks for correct answer; 2 marks for clear explanation.
5. Explain why it is easier for us to walk up a hill along a zigzag path than a straight one.
5 marks: 3 marks for correct answer; 2 marks for clear explanation.