Mathematics Ii

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  • Topic: Trigonometry, Trigonometric functions, Periodic function
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  • Published : February 6, 2013
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REVISION CHAPTER 1 (from Mid-Semester Exam Sem. II 09/10) 1

 2  Given the function : y  2 sin  2x   3  a) Find the i) amplitude ii) period iii) phase shift.

.

b) Sketch the graph of the function over one period.

[6]

2

Find the exact value of the expressions below. Rationalize the denominator where appropriate: a)

cot 70   tan 650  csc( 250  ) sec( 110  )   5   19  tan  cos   6   6    4    23  cot   sin    3   6 

[5]

b)

[5]

3

a)

From the graph of a sine or cosine curve below,find the equation of the graph in the form of:

(i) (ii)

where where

and and

. .

8

-8

.

[4]

REVISION CHAPTER 2 (from Mid-Semester Exam Sem. II 09/10)

1

Find the exact value of the expressions below. Rationalize the denominator where appropriate: a) b) tan 82.5 tan 52.5
cos 255  cos 195

[6]

2

a) Establish the identity : b)

cot   cot 2  csc 2

[5] [4]

If 2sin(x – y) = sin(x + y), prove that tanx = 3tany Given cos10o = p, express the following in terms of p: (i) (ii) (ii) tan10o tan20o cos80o [6]

3

a)

b) 4

Find the exact value of :

   4  cos  tan 1  1  cos 1    5  

[5]

Solve the equation: a)
cos 4  cos 2

Give a general formula for all the solutions. b) 3 cos   4 sin   2 ; 0    360

[6] [7]

5

a) Show that if

then

.

[6]

b)

Prove

.

[4]

6

a)

Given; , , Find the exact value of : (i) Rationalize the denominator where appropriate. [5] .

(ii)

[4]

b) 7

Find the exact value of:

[5]

Solve the equation:

a)

1  sin x cos x  4 cos x 1  sin x

;

0  x  4

[6]

b)

Give the general formula for all the solutions.

[7]

QUESTIONS FROM FINAL MATH 1 FOR CHAPTER 1 AND 2 MATHS 2 1. Without using a calculator, find the exact value of 2. If tan  4 , find the exact value of i) sec2  3. If sin t  ii) cot 

sin 700  tan  700 0 cos  430









  iii) cot     2 

8 and the terminal point for t is in quadrant IV , find csc t  sec t 17

  4. A sinusoidal function is given as y  3 cos 2 x   . What is the phase shift? 2   3  5. What is the exact value of sin 1   2   

1  6. Given cos   ,     , find the value of each of the remaining trigonometric 3 2 functions. 4 12 7. If sin    and cos   ,  in quadrant III and  in quadrant IV, what is the 5 13 value of sin     ? 8. Show that: i) sin cot   tan   sec ii) tan2   sin 2   tan2  sin 2 

iii) cot     

cot  cot   1 vi) 1  cos 2  cos 4  cos 6  2sin 3 sin 3  sin   cot   cot 

9. Solve cos 2  5 cos  3  0 , 0    2 10. Find the exact value of a. sec2 520  tan2 520 b. sec 350csc550 –tan350cot550 cos 200 0 c. cot 20  sin 200 11. If sin   0.3 , find the value of sin  sin  2   sin  4  1 12. If f    cos and f a   then find the exact value of f a   f a  2   f a  2  4 13. Find the exact value of each of the remaining trigonometric function of  if 2 3 sin    ,     3 2 14. Establish each identity sin  1  cos i)   2 csc 1  cos sin  iii) cos     cot   tan  sin  cos 

ii)

cot  tan    1  tan   cot  1  tan  1  cot 

iv) sin     sin      sin 2   sin 2 

15. Solve the equation on the interval 0    2 , tan 2  2 cos  0 16. Find the exact value of the expression:     4   4 (i) cos tan 1  1  cos 1    (ii) cos  2 tan 1     sin 1 1  3  5     17. If x  2 tan , express cos 2 as a function of x 4  18. Find the exact value of : cos 2 tan 1  3  19. Solve the following equation on the interval 0    2 : sin  cos   2 1 sin 3   cos 3  20. Establish the following identity : 1  sin 2  2 sin   cos 1 1 21. Given that cos( A  B)  , sin B   , tan A  p where p > 0 . Angle A and B 10 10 are in the...
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