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Past HSC - Trig Ratios, Sine and Cosine Rules to 2022
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ADVANCED MATHEMATICS
Trigonometry (Adv), T1 Trigonometry and Measure of Angles (Adv)
Trig Ratios, Sine and Cosine Rules (Y11)
Teacher: Toni Smith
Exam Equivalent Time: 96 minutes (based on allocation of 1 minutes per mark)
HISTORICAL CONTRIBUTION
T1 Trigonometry and Measure of Angles has contributed a healthy average of 6% per Advanced
exam since the new syllabus was introduced in 2020.
This topic has been split into four sub-topics for analysis purposes: 1-Trig Ratios, Sine and Cosine
Rules (3%), 2- 3D Trigonometry (0%), 3-Bearings (1%) and 4-Circular Measure (1%).
This analysis looks at the largest sub-topic Trig Ratios, Sine and Cosine Rules.
HSC ANALYSIS - What to expect and common pitfalls
Trig Ratios, Sine and Cosine Rules is most commonly examined using non-right angled trigonometry,
often involving "2-triangle" examples.
It has been examined via dedicated questions 8 times in the last decade (most recently in 2021) in
questions of varying difficulty, producing sub-50% mean marks on 3 occasions.
This area presents a great opportunity for high scoring, with the 2021 exam allocating 5 very
achievable marks over 2 questions.
Using one exact trig ratio to find others for further calculations has caused problems in the past and
should be reviewed (see 2015 Adv 13a).
The specific syllabus mention of the "ambiguous case" warrants attention and 2021 Q18 along with
T1 EQ-Bank 2 should be reviewed.
1. Trigonometry, 2ADV T1 2016 HSC 1 MC
2. Trigonometry, 2ADV T1 2013 HSC 2 MC
Questions
For the angle and
Which diagram best shows the angle
The diagram shows the line.
What is the slope of the line?
(A)
(B)
(C)
(D)
3. Trigonometry, 2ADV T1 2019 HSC 11a
4. Trigonometry, 2ADV T1 2021 HSC 12
5. Trigonometry, 2ADV T1 2006 HSC 1d
Using the sine rule, find the value of correct to one decimal place. (2 marks)
A right-angled triangle is cut out from a semicircle with centre. The length of the diameter
is 16 cm and = 30°, as shown on the diagram.
16 cm
X
30°
O Z
Y
NOT TO
SCALE
a. Find the length of in centimetres, correct to two decimal places. (2 marks)
b. Hence, find the area of the shaded region in square centimetres, correct to one decimal place. (
marks)
Find the value of in the diagram. Give your answer to the nearest degree. (2 marks)
6. Trigonometry, 2ADV T1 2016 HSC 12c
7. Trigonometry, 2ADV T1 2012 HSC 13a
Square tiles of side length 20 cm are being used to tile a bathroom.
The tiler needs to drill a hole in one of the tiles at a point which is 8 cm from one corner and 15 cm
from an adjacent corner.
To locate the point the tiler needs to know the size of the angle shown in the diagram.
Find the size of the angle to the nearest degree. (3 marks)
The diagram shows a triangle. The line meets the and axes at the points
and respectively. The point has coordinates.
i. Calculate the distance. (2 marks)
ii. It is known that and (Do NOT prove this)
Calculate the size of to the nearest degree. (2 marks)
iii. The point lies on such that is perpendicular to.
Find the coordinates of. (3 marks)
12. Trigonometry, 2ADV T1 2021 HSC 18
13. Trigonometry, 2ADV T1 2015 HSC 13a
The diagram shows a triangle where = 25 cm, = 16 cm, = 28° and angle
is obtuse.
Find the size of the obtuse angle correct to the nearest degree. (3 marks)
The diagram shows with sides cm, cm and cm.
i. Show that. (1 mark)
ii. By finding the exact value of , determine the exact value of the area of. (2 marks)
14. Trigonometry, 2ADV T1 2007 HSC 4c
An advertising logo is formed from two circles, which intersect as shown in the diagram.
The circles intersect at and and have centres at and.
The radius of the circle centred at is 1 metre and the radius of the circle centred at is metres.
The length of is 2 metres.
i. Use Pythagoras’ theorem to show that. (1 mark)
ii. Find and. (2 marks)
iii. Find the area of the quadrilateral. (1 mark)
iv. Find the area of the major sector. (1 mark)
v. Find the total area of the logo (the sum of all the shaded areas). (2 marks)
15. Trigonometry, 2ADV T1 2009 HSC 5c
16. Trigonometry, 2ADV T1 2018 HSC 14a
The diagram shows a circle with centre and radius 2 centimetres. The points and lie on the
circumference of the circle and.
i. There are two possible values of for which the area of is square centimetres. One
value is.
Find the other value. (2 marks)
ii. Suppose that.
(1) Find the area of sector (1 mark)
(2) Find the exact length of the perimeter of the minor segment bounded by the chord and the
arc. (2 marks)
In has length 3, has length 6 and is 60°. The point is chosen on side
so that bisects. The length is.
i. Find the exact value of the area of. (1 mark)
ii. Hence, or otherwise, find the exact value of. (2 marks)
17. Trigonometry, 2ADV T1 2013 HSC 14c
18. Trigonometry, 2ADV T1 2005 HSC 9b
The right-angled triangle has hypotenuse. The point is on such that
, and.
Using the sine rule, or otherwise, find the exact value of. (3 marks)
The triangle has a right angle at and. The line is drawn
perpendicular to. The line is then drawn perpendicular to. This process continues
indefinitely as shown in the diagram.
i. Find the length of the interval , and hence show that the length of the interval is.
(2 marks)
ii. Show that the limiting sum
is given by. (3 marks)
Copyright © 2004-22 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW)
6. Trigonometry, 2ADV T1 2016 HSC 12c
α
7. Trigonometry, 2ADV T1 2012 HSC 13a
i.
ii.
iii.
½
½
MARKER’S COMMENT: Many
students could not find the correct
equation on because they
took its gradient to be the
reciprocal of and not the
negative reciprocal.
½ 8. Trigonometry, 2ADV T1 2006 HSC 4a
i.
π
ii.
iii.
²
12. Trigonometry, 2ADV T1 2021 HSC 18
♦ Mean mark 40%.
13. Trigonometry, 2ADV T1 2015 HSC 13a
i.
ii.
²
14. Trigonometry, 2ADV T1 2007 HSC 4c
i.
ii.
iii.
²
iv.
♦ Mean mark 37%.
16. Trigonometry, 2ADV T1 2018 HSC 14a
i.
²
ii.
♦ Mean mark 36%.
STRATEGY TIP: The hint to use the
sine rule should flag to students
that they will be dealing in non-
right angled trig (i. ) and
to direct their energies at initially
finding and.
17. Trigonometry, 2ADV T1 2013 HSC 14c
18. Trigonometry, 2ADV T1 2005 HSC 9b
i.
ii.
Copyright © 2016-2023 M2 Mathematics Pty Ltd (SmarterMaths.com)
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Past HSC - Trig Ratios, Sine and Cosine Rules to 2022
Subject: math advanced
513 Documents
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ADVANCED MATHEMATICS
Trigonometry (Adv), T1 Trigonometry and Measure of Angles (Adv)
Trig Ratios, Sine and Cosine Rules (Y11)
Teacher: Toni Smith
Exam Equivalent Time: 96 minutes (based on allocation of 1.5 minutes per mark)
HISTORICAL CONTRIBUTION
T1 Trigonometry and Measure of Angles has contributed a healthy average of 6.7% per Advanced
exam since the new syllabus was introduced in 2020.
This topic has been split into four sub-topics for analysis purposes: 1-Trig Ratios, Sine and Cosine
Rules (3.3%), 2- 3D Trigonometry (0.3%), 3-Bearings (1.8%) and 4-Circular Measure (1.3%).
This analysis looks at the largest sub-topic Trig Ratios, Sine and Cosine Rules.
HSC ANALYSIS - What toexpect and commonpitfalls
Trig Ratios, Sine and Cosine Rules is most commonly examined using non-right angled trigonometry,
often involving "2-triangle" examples.
It has been examined via dedicated questions 8 times in the last decade (most recently in 2021) in
questions of varying difficulty, producing sub-50% mean marks on 3 occasions.
This area presents a great opportunity for high scoring, with the 2021 exam allocating 5 very
achievable marks over 2 questions.
Using one exact trig ratio to find others for further calculations has caused problems in the past and
should be reviewed (see 2015 Adv 13a).
The specific syllabus mention of the "ambiguous case" warrants attention and 2021 Q18 along with
T1 EQ-Bank 2 should be reviewed.
1.Trigonometry, 2ADV T1 2016 HSC 1 MC
2.Trigonometry, 2ADV T1 2013 HSC 2 MC
Questions
For the angle and
Which diagram best shows the angle
The diagram shows the line .
What is the slope of the line ?
(A)
(B)
(C)
(D)
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