Cambridge International AS and A Level Mathematics Pure Mathematics 1

Answers

P1^

  4  (i) 89 3

  5  (i)  4 d

  6  (i) BX =  33

Activity 7.1 (Page 223)

●?^ (Page^ 227)

  1 The oscillations continue to the

left.

  2 y = sin θ:

− reflect in θ = 90° to give the

curve for 90°  θ  180°

− rotate the curve for

0  θ  180° through 180°,

centre (180°, 0) to give the

curve for 180°  θ  360°.

y = cos θ:

− translate −°^900  and reflect

in y axis to give the curve for

0  θ  90°

− rotate this through 180°,

centre (90°, 0) to give the

curve for 90°  θ  180°

− reflect the curve for

0  θ  180° in θ = 180°

to give the curve for

180°  θ  360°.

Activity 7.2 (Page 228)

●?^ (Page^ 232)

The tangent graph repeats every

180° so, to find more solutions, keep

adding or subtracting 180°.

Exercise 7C (Page 233)

  1 (i), (ii)

(ii) 30°, 150°

(iii) 30°, 150° (± multiples of 360°)

(iv) −0.5

  2 (i), (ii)

(ii) x = −53°, 53°, 307°, 413°

(to nearest 1°)

(iii), (iv)

(iv)^ x^ = 53°, 127°, 413°^

(to nearest 1°)

(v) For 0  x  90°,

sin x = 0.8 and cos x = 0.6

have the same root.

For 90°  x  360°,

sin x and cos x are never

both positive.

  3 (Where relevant, answers are to

the nearest degree.)

(i) 45°, 225°

(ii) 60°, 300°

(iii) 240°, 300°

(iv) 135°, 315°

(v) 154°, 206°

(vi) 78°, 282°

(vii) 194°, 346°

(viii) 180°

  4 (i) 23

(ii)^

1

2

(iii) 1

(iv) (^12)
(v) – 21
(vi) 0
Only sin θ positive
Only tan θ positive
All positive
Only cos θ positive
θ
y
0 
y VLn θ
–180
θ
y
0 
y FRV θ
–270
–360
–90
y = cos θ
θ
y
–90 0  90  180  270  360  450 
1
–1
y = tan θ
θ
y
–90 0  90  180  270  360  450 
y = sin θ
θ
y
–90 0  90  180  270  360  450 
1
–1
x
VLQ x
30 0 20

 2
±
0
0
360 
x
FRV x
–90–3 3 307 13
1
–1
90  180  270  360 0
06
x
VLn x
–90 3 127  13
1
–1
90  180  270  360 0
08