Date | May 2009 | Marks available | 5 | Reference code | 09M.1.hl.TZ1.5 |
Level | HL only | Paper | 1 | Time zone | TZ1 |
Command term | Find, Show that, and Hence or otherwise | Question number | 5 | Adapted from | N/A |
Question
(a) Show that arctan(12)+arctan(13)=π4 .
(b) Hence, or otherwise, find the value of arctan(2)+arctan(3) .
Markscheme
(a) METHOD 1
let x=arctan12⇒tanx=12 and y=arctan13⇒tany=13
tan(x+y)=tanx+tany1−tanxtany=12+131−12×13=1 M1
so, x+y=arctan1=π4 A1AG
METHOD 2
for x, y>0 , arctanx+arctany=arctan(x+y1−xy) if xy<1 M1
so, arctan12+arctan13=arctan(12+131−12×13)=π4 A1AG
METHOD 3
an appropriate sketch M1
e.g.
correct reasoning leading to π4 R1AG
(b) METHOD 1
arctan(2)+arctan(3)=π2−arctan(12)+π2−arctan(13) (M1)
=π−(arctan(12)+arctan(13)) (A1)
Note: Only one of the previous two marks may be implied.
=π−π4=3π4 A1 N1
METHOD 2
let x=arctan2⇒tanx=2 and y=arctan3⇒tany=3
tan(x+y)=tanx+tany1−tanxtany=2+31−2×3=−1 (M1)
as π4<x<π2(accept 0<x<π2)
and π4<y<π2(accept 0<y<π2)
π2<x+y<π(accept 0<x+y<π) (R1)
Note: Only one of the previous two marks may be implied.
so, x+y=3π4 A1 N1
METHOD 3
for x, y>0 , arctanx+arctany=arctan(x+y1−xy)+π if xy>1 (M1)
so, arctan2+arctan3=arctan(2+31−2×3)+π (A1)
Note: Only one of the previous two marks may be implied.
=3π4 A1 N1
METHOD 4
an appropriate sketch M1
e.g.
correct reasoning leading to 3π4 R1A1
[5 marks]
Examiners report
Most candidates had difficulties with this question due to a number of misconceptions, including arctanx=tan−1x=cosxsinx and arctanx=arcsinxarccosx, showing that, although candidates were familiar with the notation, they did not understand its meaning. Part (a) was done well among candidates who recognized arctan as the inverse of the tangent function but just a few were able to identify the relationship between parts (a) and (b). Very few candidates attempted a geometrical approach to this question.