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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=arctan12tanx=12 and y=arctan13tany=13

tan(x+y)=tanx+tany1tanxtany=12+13112×13=1     M1

so, x+y=arctan1=π4     A1AG

METHOD 2

for x, y>0 , arctanx+arctany=arctan(x+y1xy) if xy<1     M1

so, arctan12+arctan13=arctan(12+13112×13)=π4     A1AG

METHOD 3

an appropriate sketch     M1

e.g.    

correct reasoning leading to π4     R1AG

 

(b)     METHOD 1

arctan(2)+arctan(3)=π2arctan(12)+π2arctan(13)     (M1)

=π(arctan(12)+arctan(13))     (A1)

Note: Only one of the previous two marks may be implied.

 

=ππ4=3π4     A1     N1

METHOD 2

let x=arctan2tanx=2 and y=arctan3tany=3

tan(x+y)=tanx+tany1tanxtany=2+312×3=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+y1xy)+π if xy>1     (M1)

so, arctan2+arctan3=arctan(2+312×3)+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=tan1x=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.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.5 » The inverse functions xarcsinx , xarccosx , xarctanx ; their domains and ranges; their graphs.

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