HSC Exam Questions (Trigonometric Ratios) Note to students: For those who have yet to study radians, 2 = 360°. 1995 HSC Q3(b) A horizontal bridge is built between points and . The bridge is supported by cables and , which are attached to the top of a vertical pylon . 4 The section of the pylon, , above the bridge is 8 metres long and ∠ = 52°. (i) Find the length of the cable . (ii) Find the size of ∠ to the nearest degree. 1997 HSC Q7(a) By expressing sec and tan in terms of sin and cos , show that 2 sec ! − tan ! = 1 1998 HSC Q1(e) Find the exact value of sin ! ! + sin !! ! . 2 1999 HSC Q3(c) In the diagram, is parallel to , is 6 cm, is 3 cm and is 7 cm. 4 (i) Use the cosine rule to find the size of ∠, to the nearest degree. (ii) Hence find the size of ∠, giving reasons for your answer.
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HSC Exam Questions (Trigonometric Ratios) Note to students: For those who have yet to study radians, 2𝜋 = 360°. 1995 HSC Q3(b)
A horizontal bridge is built between points 𝐴 and 𝐵. The bridge is supported by cables
𝑆𝑃 and 𝑆𝑅, which are attached to the top of a vertical pylon 𝑆𝑇. 4
The section of the pylon, 𝑆𝑄, above the bridge is 8 metres long and ∠𝑆𝑅𝑄 = 52°.
(i) Find the length of the cable 𝑆𝑅.
(ii) Find the size of ∠𝑆𝑃𝑄 to the nearest degree.
1997 HSC Q7(a)
By expressing sec𝜃 and tan𝜃 in terms of sin𝜃 and cos𝜃, show that 2
sec! 𝜃 − tan! 𝜃 = 1
1998 HSC Q1(e)
Find the exact value of sin !!+ sin !!
!. 2
1999 HSC Q3(c)
In the diagram, 𝐴𝐶 is parallel to 𝐷𝐵, 𝐴𝐵 is 6 cm, 𝐵𝐶 is 3 cm and 𝐴𝐶 is 7 cm. 4
(i) Use the cosine rule to find the size of ∠𝐴𝐶𝐵, to the nearest degree.
(ii) Hence find the size of ∠𝐷𝐵𝐶, giving reasons for your answer.
2000 HSC Q1(c)
What is the exact value of cos !!? 1
2000 HSC Q5(a)
Solve tan 𝑥 = 2 for 0 < 𝑥 < 2𝜋. 2
Express your answer in radian measure correct to two decimal places.
2000 HSC Q9(c)
The diagram shows a square 𝐴𝐵𝐶𝐷 of side 𝑥 cm, with a point 𝑃 within the square, such
that 𝑃𝐶 = 6 cm, 𝑃𝐵 = 2 cm and 𝐴𝑃 = 2 5 cm. 7
Let ∠𝑃𝐵𝐶 = 𝛼.
(i) Using the cosine rule in triangle 𝑃𝐵𝐶, show that cos𝛼 = !!!!"!!
.
(ii) By considering triangle 𝑃𝐵𝐴, show that sin𝛼 = !!!!"!!
.
(iii) Hence, or otherwise, show that the value of 𝑥 is a solution of
𝑥! − 56𝑥! + 640 = 0
(iv) Find 𝑥. Give reasons for your answer.
2001 HSC Q3(d)
The diagram shows a triangle with sides 7 cm, 13 cm and 𝑥 cm, and an angle of 60° as
marked.
Use the cosine rule to show that 𝑥! − 7𝑥 = 120, and hence state the exact value of 𝑥. 4
2002 HSC Q2(c)
In the diagram, 𝑋𝑌𝑍 is a triangle where ∠𝑍𝑌𝑋 = 45° and ∠𝑍𝑋𝑌 = 60°.
Find the exact value of the ratio !!. 3
2002 HSC Q4(b)
Find all values of 𝜃, where 0° ≤ 𝜃 ≤ 360°, that satisfy the equation: 2
cos𝜃 −25 = 0
Give your answer(s) to the nearest degree.
2002 HSC Q4(c)
In the diagram, 𝐿𝑀𝑁 is a triangle where 𝐿𝑀 = 5.2 metres, 𝐿𝑁 = 8.9 metres and angle
𝑀𝐿𝑁 = 110°.
(i) Find the length of 𝑀𝑁. 2
(ii) Calculate the area of triangle 𝐿𝑀𝑁. 2
2003 HSC Q4(a)
In the diagram, the point 𝑄 is due east of 𝑃. The point 𝑅 is 38 km from 𝑃 and 20 km from
𝑄. The bearing of 𝑅 from 𝑄 is 325°.
(i) What is the size of ∠𝑃𝑄𝑅? 1
(ii) What is the bearing of 𝑅 from 𝑃? 3
2004 HSC Q9(a)
Solve 2 sin! 𝑥 − 3 sin 𝑥 − 2 = 0 for 0 ≤ 𝑥 ≤ 2𝜋. 2
2004 HSC Q3(c)
The diagram shows a point 𝑃 which is 30 km due west of the point 𝑄.
The point 𝑅 is 12 km from 𝑃 and has a bearing from 𝑃 of 070°.
(i) Find the distance of 𝑅 from 𝑄. 2
(ii) Find the bearing of 𝑅 from 𝑄. 2
2004 HSC Q8(a)
(i) Show that cos𝜃 tan𝜃 = sin𝜃. 1
(ii) Hence solve 8 sin𝜃 cos𝜃 tan𝜃 = cosec𝜃 for 0 ≤ 𝜃 ≤ 2𝜋. 2
2005 HSC Q2(a)
Solve cos𝜃 = !! for 0 ≤ 𝜃 ≤ 2𝜋. 2
2005 HSC Q3(b)
The lengths of the sides of a triangle are 7 cm, 8 cm and 13 cm.
(iii) Find the size of the angle opposite the longest side. 2
(iv) Find the area of the triangle. 1
2006 HSC Q1(d)
Find the value of 𝜃 in the diagram. Give your answer to the nearest degree. 2
2007 HSC Q4(a)
Solve 2 sin 𝑥 = 1 for 0 ≤ 𝑥 ≤ 2𝜋. 2
2008 HSC Q6(a)
Solve 2 sin! !!= 1 for –𝜋 ≤ 𝑥 ≤ 𝜋. 3
2009 HSC Q1(d)
Find the exact value of 𝜃 such that 2 cos𝜃 = 1, where 0 ≤ 𝜃 ≤ !!. 2
2010 HSC Q5(b)
(i) Prove that sec! 𝑥 + sec 𝑥 tan 𝑥 = !!!"#!!"#! !
. 1
(ii) Hence prove that 1
sec! 𝑥 + sec 𝑥 tan 𝑥 =1
1− sin 𝑥
(iii) Hence use the table of standard integrals to find the exact value of 2
11− sin 𝑥
!!
!𝑑𝑥
2011 HSC Q2(b)
Find the exact values of 𝑥 such that 2 sin 𝑥 = − 3, where 0 ≤ 𝑥 ≤ 2𝜋. 2
2011 HSC Q8(a)
In the diagram, the shop at 𝑆 is 20 kilometres across the bay from the post office at 𝑃.
The distance from the shop to the lighthouse at 𝐿 is 22 kilometres and ∠𝑆𝑃𝐿 is 60°.
Let the distance 𝑃𝐿 be 𝑥 kilometres.
(i) Use the cosine rule to show that 𝑥! − 20𝑥 − 84 = 0. 1
(ii) Hence, find the distance from the post office to the lighthouse. Give your answer
to the nearest kilometre. 2
2012 HSC Q6
What are the solutions of 3 tan 𝑥 = −1 for 0 ≤ 𝑥 ≤ 2𝜋?
(A) !!! and !!
!
(B) !!! and !!
!
(C) !!! and !!
!
(D) !!! and !!!
!
2013 HSC Q14(c)
The right-‐angled triangle 𝐴𝐵𝐶 has hypotenuse 𝐴𝐵 = 13. The point 𝐷 is on 𝐴𝐶 such that
𝐷𝐶 = 4, ∠𝐷𝐵𝐶 = !! and ∠𝐴𝐵𝐷 = 𝑥.
Use the sine rule, or otherwise, find the exact value of sin 𝑥. 3
2014 HSC Q7
How many solutions of the equation sin 𝑥 − 1 tan 𝑥 + 2 = 0 lie between 0 and 2𝜋?
(A) 1
(B) 2
(C) 3
(D) 4
2014 HSC Q13(d)
Chris leaves island 𝐴 in a boat and sails 142 km on a bearing of 078° to island 𝐵. Chris
then sails on a bearing of 191° for 220 km to island 𝐶, as shown in the diagram.
(i) Show that the distance from island 𝐶 to island 𝐴 is approximately 210 km. 2
(ii) Chris wants to sail from island 𝐶 directly to island 𝐴. On what bearing should
Chris sail? Give your answer correct to the nearest degree. 3
2014 HSC Q15(a)
Find all solutions of 2 sin! 𝑥 + cos 𝑥 − 2 = 0, where 0 ≤ 𝑥 ≤ 2𝜋. 3
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