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National Exemplar Paper 2
TRIGONOMETRY [52]
QUESTION 6
QUESTION 7
QUESTION 5
5.1 In the figure alongside,
the point P(- 5; b) is
P(- 5; b)
plotted on the
Cartesian plane.
13
In the diagram below, the graphs of f(x) = cos(x + p)
and g(x) = q sin x are shown for the interval
-180º ≤ x ≤ 180º.
7.1 Prove that in any acute-angled ΔABC,
y
sin A
= sin C .
a
c
(5)
y
α
O
OP = 13 units and
ˆ = α.
ROP
Q
R
x
7.2 In ΔPQR, P̂ = 132º, PQ = 27,2 cm and
QR = 73,2 cm.
1
g
A
2
P
f
0,5
132º
27,2 cm
5.1.1 cos α
-180º -135º - 90º - 45º 0º
(1)(3)
sin(θ - 360º) sin(90º - θ) tan(- θ)
cos(90º + θ)
to a single trigonometric ratio.
(5)
5.2.2 Hence, or otherwise, without using a
calculator, solve for θ if 0º ≤ θ ≤ 360º :
5.3.1 Prove that
8
4
4
=
.
1 - cos A
sin2 A 1 + cos A
5.3.2 For which value(s) of A in the interval
0º ≤ A ≤ 360º is the identity in
QUESTION 5.3.1 undefined?
5.4
Determine the general solution of
2
8 cos x - 2 cos x - 1 = 0.
135º 180º
x
- 0,5
(3)
(5)
R
73,2 cm
B
-1
sin(θ - 360º) sin(90º - θ) tan(- θ)
5.2 Consider :
cos(90º + θ)
sin(θ - 360º) sin(90º - θ) tan(- θ)
= 0,5
cos(90º + θ)
90º
Q
5.1.2 tan(180º - α)
5.2.1 Simplify
45º
7.2.1 Calculate the size of R̂.
(3)
7.2.2 Calculate the area of ΔPQR.
(3)
6.1 Determine the values of p and q.
(2)
6.2 The graphs intersect at A(- 22,5º ; 0,38) and B.
Determine the coordinates of B.
ˆ = b and
ˆ = a, PQS
7.3 In the figure below, SPQ
PQ = h. PQ and SR are perpendicular to RQ.
(2)
P
a
6.3 Determine the value(s) of x in the interval
-180º ≤ x ≤ 180º for which f(x) - g(x) < 0.
(2)
S
h
6.4 The graph f is shifted 30º to the left to obtain
a new graph h.
6.4.1 Write down the equation of h in its
simplest form.
b
R
6.4.2 Write down the value of x for which h
has a minimum in the interval
(1) [9]
-180º ≤ x ≤ 180º.
(3)
Q
(2)
7.3.1 Determine the distance SQ in terms of
a, b and h.
7.3.2 Hence show that RS =
(3)
h sin a . cos b
. (3) [17]
sin(a + b)
(6) [26]
Q5
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EXAM PAPERS: PAPER 2
Without using a calculator, determine the
value of the following :