IB Mathematics Higher Level Sample Questions Trig Answers

MATHEMATICS HL  – PAPER ONE STYLE QUESTIONS Circular Functions and Trigonometry

Question 1. (a) Method 1:

sicosn = 34 416sisinn= 3=co9coss      16co   = 25co  16si n  16cos s 25 cos s 1625 = cos  cos = 45 0 ≤  ≤ 2    =     =    =    sin =  sin2 = 2sin cos−= 2×2 × 35 × 45 = 2254 tan   =   =  =   =  Method 2:

(b)

(c)

−   



Question 2.

0 <  ≤  0 < sin ≤ 1 ∴ sin  ≤ sisin ≤ si1n si n  =  1    =  −−   ×  ×180 = 36°= = cos270°cos30°sin270°sin30° cos30° cos300°cos30° = 10sin30°cos30° =2−sin60° = √4 3     + ×tan      = − × tan    =   For

,

Question 3. (a)

(b)

(c)

(d)

Question 4. (a)

−  = − = −  +− − +− + − − + 2 sin  = 8

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sin  = 141 si n=30°= ±, 1250°, 210°,330° Page | 2 Question 5. (a)

 = 0 cos sisinnsin = 12si n cos = 11× 1  = 0 si n=0,=0, ,  , 2 sin  = 1    sisinn=si1n = 2  =  sisinn  sisinn 2= 2 = 0 sin 2sin 1 = 0

(b) (i)

or

(ii)

Alternative Method:

Question 6. (a) Period =

(b)

 = 4 

 == 4 2si=n2 = √ 3   √ 3 si n 2  =22 2 == 3 ,, 3 63

Question 7.

 = 5 22 = 32  = 5 22 = 72 2 =  =2

Question 8. (a)

2 ≤= ≤ 2=  +   = + = 

(b) Min

(c) Max

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Question 9. (a)

(b)

2cos1sin  = 0 cos =30  sin = 1 cos= 22si , 2 n cos = 0 cos12sin  = 0  =  ,  ,  , 

(4 solutions)

Question 10. (a) (b)

AE2=sin2si =n9 28  = 9 = =636

Question 11. (a)

 sin si√ 2n  cos cos 2 √ 25  √ 5 = √ 2√ 25 = √ 110

Question 12.

   = =         =     21   525  =   64 =  25= 64 = 5 −  = 27

(a) (i)

(ii)

(iii) (b)

(c)

14  = 34 ×27 = =981 Page | 4

Question 13.

 °

 = °  = 2sisinn30°120° 3 √  2×  = 12 2  = 2√ 3 Question 14.

tan = CX = X+  = √ + = √ + = √ − Question 15.

 = 6  2  2×6×2×   = 3649  = 31

Question 16.

 42 = 27 3 = 27  = 227

 = 3√ 26 Question 17. (a)

 = 60

=    2 =    == 2× 2 ×2×2×si =  n150° = 4×  = 2 cm  

(b) (i) Length of minor arc BC (ii) Area of sector OBC (c) Area of shaded region

Question 18. (a) (b) (c)

90 = 36 90 = 54 =  902 = 45 =   = 30

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Question 19. (a) (b)

    = − − = 1 − 4 = 5

tan= 1= 1  = 4 Question 20. (a)

(b)

4848 == 43    4  = =416  84√   = 8√ 3 cm 



Question 21.

=  =  = 8 cm   2×  ×2×8× √    ×2×2×  = 8√ 3   cm

(a) OC (b)

Question 22.

|| = − = 4  = 4 2 = 3  =   = + = 6

(Reflection in

=

)

Question 23.

 =     + | =2| =  = 2

(a) (i)

(ii)

(b)

(c) (d)

(e)

2 = 25  = 351  == 1 2 tan2 = √  2 =6  =≤ 12

 =    2  =    2  =  ,  ,  ,  sum =  =  Or

Question 24. (a)

1 = 304cos1 1  = 304  2 1 = 28 dB

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(b) 26 dB (c)

  =  cos 43  = 6  = 18      =  =  = 

(d) Period



Question 25. (a)

  °3000 2. 5 = 950cos36×2. 5 2.5 = 3000

(b) 2050 (c)

(d)

   =   years

Question 26.

2sincos  √ 3 cos = 0 cos(2sin  √ 3) = 0 cos = 0  sin = √   =  ,  , 

Question 27.

√ 2 cos × √  sin  × √   = cos sin  coscos si n cos si n   = 0 cos sincos 1 = 0 (a)

(b)

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√ 2 cos 4cos 1 = 0 cos 3 4 = 0 or cos = 1  = 4 , Question 28.

sin 2A = 2si√ 2nAcosA sin 2A = 2× 4 ×2√ √ 72 = √4 7   cos2B = 2cos B1 = 2× 1 =    √  × √   √  × √  = √ +√ 

(a) (i)

(ii) (b)

Question 29.

1cos  2cos 6 = 8cos  2cos 1 = 81cos 1

(a) (i) 6 (ii) (b) 8

Question 30. (a)

(b)

  15si n  1si n  = 8si n  2si2n  1515si n  7 = 0 7 = 0 2 1 1 7 = 0 sin =52  = 6, 6

Question 31.

4sinsin  1sin 1 = 0 4sin  1si n  1  = 0 sin  = ±  or sin  = 1  =  ,  ,  ,  , 

Question 32.

sin = 2sin cos 2cossin tan = 2sin  2costan tan12cos = 2sin

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   tan = +  = 2,  = 2 Question 33.

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cos3 = cos2cos si n 2si n    = = 2cos2cos cos 1cos2cos1cos 2cossin   =cos90°° 4cos  3cos    = cos90°cos°si n 90°si n ° = 0sin°

(a) (i)

(b)

Question 34. (a) (b)

−   ++ −    +       =   =  = si n     +  +  +  +   +  −+ = +  cos  = 1 cos = 2  1

Question 35.

cos  = 0 cos  = √  {  ==    { ==  

{ ==   { ==   

{  ==    =   = 

 {==   =  = 

Question 36.

 sec  = 1 2tan tan   sec  = 12tan tan      sec  = 21tan       = 2

Question 37. (a) (b)

 ×2 ×  =   cm  4sin  =   44 1 

 22 = 26 = 3  cm from O

Question 38. Draw perpendicular CH to AD.

AD = AHHD HD = cos AH = sin2  = sin   = cos sin  Question 39.

22√ 3  1 0 √  +− cos = 6 √  cos = 8√ 3  6      sin = 1 8√ 3 sin = 6√ 2413

(a) (i)

(ii)

(iii) (b)

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Question 40.

1 ≤  ≤ 3

(a) (i) (ii)

Page | 11

(b)

(c)

 = 2sin   1  1= 2sin   21 2   = =sisinn− 12 2 2    −  −  = sin   

Question 41. (a) (i)

Page | 12

(ii)

(iii) (b)



 = arccos cos =   sin  =    = arctan tan =   =  sin = √  cos = √  45 sinarccos45arctan == sisinnarccos45arctan45 = 3sincos cossi n  = 5 × √ 541  45 × √ 441 = 5√ 141 = √205 41