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Higher Mathematics
Trigonometry
Paper 1 Section B
1. (a) Solve the equation sinPSfrag
2x ◦ − cos
x ◦ = 0 in the interval 0 ≤ x ≤ 180.
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y
y = sin 2x ◦
(b) The diagram shows parts of two
trigonometric graphs, y = sin 2x ◦
and y = cos x ◦ .
180
x
90
O
Use your solutions in (a) to write
down the coordinates of the point P.
y = cos x ◦
P
[SQA]
Part
(a)
(b)
•1
•2
•3
•4
Marks
4
1
ss:
pd:
pd:
pd:
•5 ic:
Level
C
C
Calc.
NC
NC
Content
T10
T3
Answer
30, 90, 150
√
(150, − 23 )
use double angle formula
factorise
process
process
or
interpret graph
•1
•2
•3
•4
2 sin x ◦ cos x ◦
cos x ◦ (2 sin x ◦ − 1)
cos x ◦ = 0, sin x ◦ =
90, 30, 150
U2 OC3
2001 P1 Q5
1
2
•3 sin x ◦ = 12 and x = 30, 150
•4 cos x ◦ = 0 and x = 90
√ •5 150, − 23
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1
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Higher Mathematics
[SQA]
2. Functions f and g are defined on suitable domains by f (x) = sin(x ◦ ) and
g(x) = 2x .
(a) Find expressions for:
(i) f (g(x));
2
(ii) g( f (x)).
5
(b) Solve 2 f (g(x)) = g( f (x)) for 0 ≤ x ≤ 360.
Part
(a)
(b)
Marks
2
5
•1 ic:
•2 ic:
Level
C
C
Calc.
CN
CN
Content
A4
T10
Answer
(i) sin(2x ◦ ), (ii) 2 sin(x ◦ )
0◦ , 60◦ , 180◦ , 300◦ , 360◦
2002 P1 Q3
•1 sin(2x ◦ )
•2 2 sin(x ◦ )
interpret f (g(x))
interpret g( f (x))
•3
•4
•5
•6
ss: equate for intersection
ss: substitute for sin 2x
pd: extract a common factor
pd: solve a ‘common factor’
equation
•7 pd: solve a ‘linear’ equation
•3
•4
•5
•6
•7
2 sin(2x ◦ ) = 2 sin(x ◦ )
appearance of 2 sin(x ◦ ) cos(x ◦ )
2 sin(x ◦ ) (2 cos(x ◦ ) − 1)
sin(x ◦ ) = 0 and 0, 180, 360
cos(x ◦ ) = 12 and 60, 300
or
•6 sin(x ◦ ) = 0 and cos(x ◦ ) =
•7 0, 60, 180, 300, 360
[SQA]
U2 OC3
1
2
3.
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Higher Mathematics
5
4. Solve the equation sin 2x ◦ + sin x ◦ = 0, 0 ≤ x < 360.
[SQA]
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5. The diagram shows the graph of a
cosine function from 0 to π .
[SQA]
(a) State the equation of the graph.
√
(b) The line with equation y = − 3
intersects this graph at point A
and B.
Marks
1
3
•1 ic:
Level
C
C
Calc.
NC
NC
1
O
π
2
Content
T4
T7
interpret graph
•2 ss: equate equal parts
•3 pd: solve linear trig equation in
radians
•4 ic: interpret result
A B
Answer
y = 2 cos
√2x
7π
B( 12 , − 3)
U2 OC3
2002 P1 Q8
√
•1 2 cos 2x = − 3
7π
•2 2x = 5π
6 , 6
7π
3
• x = 12
π
4
has its maximum
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3
•1 2 cos 2x
6. Find the values of t, where 0 < t < 2π , for which 4 cos 2t −
value.
[SQA]
x
√
y=− 3
π
−2
Find the coordinates of B.
Part
(a)
(b)
y
2
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7.
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8.
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4
9. Solve 2 sin 3x ◦ − 1 = 0 for 0 ≤ x ≤ 180.
[SQA]
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10. Solve the equation 2 cos2 x = 12 , for 0 ≤ x ≤ π .
[SQA]
3
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11.
[SQA]
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12. Find the exact solutions of the equation 4 sin2 x = 1, 0 ≤ x < 2π .
[SQA]
4
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13.
[SQA]
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4
14. Solve the equation 2 sin 2x − π6 = 1, 0 ≤ x < 2π .
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15.
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16.
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Higher Mathematics
17. Given that cos D =
[SQA]
√2
5
and 0 < D <
π
2,
find the exact values of sin D and cos 2D .
3
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18. Given that sin A = 34 , where 0 < A <
[SQA]
π
2,
find the exact value of sin 2A.
3
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19. For acute angles P and Q, sin P =
[SQA]
12
13
and sin Q = 35 .
Show that the exact value of sin(P + Q) is
63
65 .
3
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20. Find the exact value of sin θ ◦ + sin(θ ◦ + 120◦ ) + cos(θ ◦ + 150◦ ).
[SQA]
3
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21. If cos θ = 45 , 0 ≤ θ <
[SQA]
π
2,
find the exact value of
(a) sin 2θ
2
(b) sin 4θ .
3
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22. Given that tan α =
[SQA]
√
11
3 ,
0<α<
π
2,
find the exact value of sin 2α.
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Higher Mathematics
23. If x ◦ is an acute angle such that tan x ◦ =
√
4 3+3
◦
◦
sin(x + 30 ) is
.
10
[SQA]
4
3,
show that the exact value of
3
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24. A and B are acute angles such that tan A =
[SQA]
3
4
and tan B =
5
12 .
Find the exact value of
(a) sin 2A
2
(b) cos 2A
1
(c) sin(2A + B).
2
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25.
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26. Functions f (x) = sin x , g(x) = cos x and h(x) = x +
set of real numbers.
[SQA]
π
4
are defined on a suitable
(a) Find expressions for:
(i) f (h(x));
2
(ii) g(h(x)).
(b) (i) Show that f (h(x)) =
√1
2
sin x +
√1
2
cos x .
(ii) Find a similar expression for g(h(x)) and hence solve the equation
f (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π .
Part
(a)
Marks
2
Level
C
Calc.
NC
(b)
5
C
NC
•1 ic:
•2 ic:
•3
•4
•5
•6
•7
ss:
ic:
ic:
pd:
pd:
Content
A4
T8, T7
interpret composite functions
interpret composite functions
Answer
(i)
sin(x + π4 ),
(ii)
cos(x + π4 )
(i) proof, (ii) x = π4 , 3π
4
•1 sin(x + π4 )
•2 cos(x + π4 )
•3 sin x cos π4
complete
•4 g(h(x)) =
expand sin(x + π4 )
interpret
substitute
start solving process
process
+
cos x sin π4
5
U2 OC3
2001 P1 Q7
and
√1 cos x − √1 sin x
2
2
•5 ( √12 sin x + √12 cos x) − ( √12 cos x − √12
•6 √22 sin x
•7 x = π4 , 3π
accept only radians
4
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sin x)
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Higher Mathematics
y
27. On the coordinate diagram shown, A is the
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point (6, 8) and B is the point (12, −5).
Angle
AOC = p and angle COB = q.
[SQA]
A(6, 8)
4
Find the exact value of sin(p + q).
O
p
C
q
x
B(12, −5)
Part
•1
•2
•3
•4
Marks
4
ss:
pd:
ic:
pd:
Level
C
Calc.
NC
Content
T9
Answer
U2 OC3
63
65
2000 P1 Q1
know to use trig expansion
•1 sin p cos q + cos p sin q
process missing sides
•2 10 and 13
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8 12
6 5
interpret data
•3 10
· 13 + 10
· 13
126
4
O
process
• 130
x
y
28. In triangle ABC, show that the exact
2
value of sin(a + b) is √ .
5
[SQA]
C
a
A
Part
•1
•2
•3
•4
Marks
4
pd:
ss:
pd:
pc:
Level
C
Calc.
NC
Content
T9
process the missing sides
expand
substitute
process and complete proof
4
1
1
3
Answer
proof
b
B
U2 OC3
2002 P1 Q5
√
√
•1 AC = 2 and BC = 10
stated or implied by •3
2
• sin(a + b) = sin a cos b + cos a sin b
•3 √12 · √310 + √12 · √110
•4
√4
20
= ... =
√2
5
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29.
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30.
[SQA]
(a) Show that (cos x + sin x)2 = 1 + sin 2x .
Z
(b) Hence find (cos x + sin x)2 dx .
1
3
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31.
[SQA]
(a) By writing sin 3x as sin(2x + x), show that sin 3x = 3 sin x − 4 sin3 x .
Z
(b) Hence find
sin3 x dx .
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[END OF PAPER 1 SECTION B]
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Paper 2
5
1. Solve the equation 3 cos 2x ◦ + cos x ◦ = −1 in the interval 0 ≤ x ≤ 360.
[SQA]
Part
•1
•2
•3
•4
•5
Marks
5
ss:
pd:
ss:
pd:
pd:
Level
A/B
Calc.
CR
Content
T10
know to use cos 2x = 2 cos2 x − 1
process
know to/and factorise quadratic
process
process
Answer
60, 131·8, 228·2, 300
•1
•2
•3
•4
•5
U2 OC3
2000 P2 Q5
3(2 cos2 x ◦ − 1)
6 cos2 x ◦ + cos x ◦ − 2 = 0
(2 cos x ◦ − 1)(3 cos x ◦ + 2)
cos x ◦ = 12 , x = 60, 30
cos x ◦ = − 23 , x = 132, 228
2. Solve the equation cos 2x ◦ + 5 cos x ◦ − 2 = 0, 0 ≤ x < 360.
[SQA]
5
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5
3. Solve the equation cos 2x ◦ + cos x ◦ = 0, 0 ≤ x < 360.
[SQA]
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4.
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5.
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6.
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7. Find, correct to one decimal place, the value of x between 180 and 270 which
satisfies the equation 3 cos(2x ◦ − 40◦ ) − 1 = 0.
[SQA]
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8. If f (a) = 6 sin2 a − cos a, express f (a) in the form p cos 2 a + q cos a + r .
[SQA]
Hence solve, correct to three decimal places, the equation 6 sin2 a − cos a = 5 for
0 ≤ a ≤ π.
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9.
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10.
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11.
[SQA]
(a) Show that 2 cos 2x ◦ − cos2 x ◦ = 1 − 3 sin2 x ◦ .
2
(b) Hence solve the equation 2 cos 2x ◦ − cos2 x ◦ = 2 sin x ◦ in the interval
0 ≤ x < 360.
4
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12.
[SQA]
(a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that,
for cos θ , it has equal roots.
3
(b) Show that there are no real roots for θ .
1
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13.
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14. The displacement,
√ d units, of a wave after t seconds, is given by the formula
d = cos 20t◦ + 3 sin 20t◦ .
[SQA]
(a) Express d in the form k cos(20t ◦ − α◦ ), where k > 0 and 0 ≤ α ≤ 360.
(b) Sketch the graph of d for 0 ≤ t ≤ 18.
(c) Find, correct to one decimal place, the values of t, 0 ≤ t ≤ 18, for which the
displacement is 1·5 units.
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4
3
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15.
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16.
[SQA]
√
(a) Show that 2 cos(x ◦ + 30◦ ) − sin x ◦ can be written as 3 cos x ◦ − 2 sin x ◦ .
√
(b) Express 3 cos x ◦ − 2 sin x ◦ in the form k cos(x ◦ + α◦ ) where k > 0 and
0 ≤ α ≤ 360 and find the values of k and α.
(c) Hence, or otherwise, solve the equation 2 cos(x ◦ + 30◦ ) = sin x ◦ + 1,
0 ≤ x ≤ 360.
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[END OF PAPER 2]
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4
3