chapter 3_extra exercise

LFSE012_Chapter3_Extra Exercise 1

EXTRA EXERCISE – CHAPTER 3 – TRIGONOMETRY

1. In each of the following express the ratio in exact form:

a) Find , given 4

tan3

and 3

2

b) Find , given 2

sin5

and 2

c) Find , given 5

tan3

and 3

22

2. a) Sketch the graph of in the interval for

b) Use this graph to solve the equation

c) Use this graph to solve the equation

3. Find, without using a calculator:

a) given 1

sin3

A b) given 2

tan5

A

4. If and are acute angles such that 4

cos5

and 8

tan15

, find

a) ( ) b) ( ) c) the quadrant containing

5. If 4

sin5

and 5

sec3

for a third-quadrant angle and a first-quadrant angle , find

a) ( ) b) ( ) c) the quadrant containing

6. Solve the following equations for :

a) 2 3

sin4

x b) c)

d) e) f) √

g) h)

7. Prove the following identity:

a) sin

cot1 cos 2

A A

A

b)

cos sinsec2 tan 2

cos sin

A AA A

A A

c) cot tan

sec2cot tan

A AA

A A

8. Evaluate the following without using a calculator (if they exist):

a) 1 3Tan tan

4

b) 1Cos sin6

c) 1 1Cos

2


9. Evaluate the following without using a calculator:

a) 5

tan Artan13

b) 1Cos cos540

c) 3

cos 2 Arsin 2

d) 1

sec Arsin3

10. Evaluate, giving answers as exact values:

a) 1 13 3sin Sin Sin

5 5

b)

5 4cos Arsin Arsin

13 5

c) 1 14 5cos Tan Cos

3 13

11. Without using a calculator, show the following:

a) 1 13 3Sin Sin 0

5 5

b)

1 13 242Sin Sin

5 25

c) 1 15 5

Tan Cos12 13 2

d) 3 3

Arcos Artan 5 4 2

12. Show that 1 1 2Cos Sin 1x x for

13. A buoy in the sea is rising and falling vertically with the waves. Its height metres above the

mean sea level after seconds from the time first observed is:

( )

a) Find this height for and

b) Draw a graph showing against .

c) From the graph, find when the height is 1.5 metres above mean sea level.

14. a) Draw the graph of for

b) Use the graph to solve the equation 3

3cos sin2

x x

15. Solve the equation , for :

16. Solve the equation for

17. Solve the equation tan 3 cot3

for 0 2

*18. Solve the equation cos sin 16 6

for 0 2


ANSWERS:

1. 3 2 21 5 34

a) b) c) 5 21 34

2. a)

b) 3

,2 2

c) 3 5 7

, , ,4 4 4 4

3. 7 20

a) b) 9 29

4. 77 36

a) b) c) quadrant I85 85

5. 24 24

a) b) c) quadrant IV25 7

6. a) { } b) { }

c) { } d) { }

e) { } f) { }

g) { } h) { }

8. i) ii) iii) 4 3 4

9. 5 1 3 2

i) ii) iii) iv) 13 2 4

10. 16 63

i) 0 ii) iii) 65 65

13. a)

b) c) and (approx.)

0 2 4 6 8 10 12

2 1 1 2 1 1 2


14. a) b) From the graph, values are approximately and

15. { }

16. { }

17. 7 13 19

, , ,12 12 12 12

18. 5

,6 3

chapter 3_extra exercise

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