# Worksheet 3 Trigonometry by vcl99353

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```									Worksheet 3 Trigonometry
Determine the amplitude and period p of each function f whose values are indicated, and
then sketch the graph over the following interval of two periods " p ! x ! p.

!x                                     &    (#
1. f(x) = sin                          2. f(x) = 2 sin \$ x ' !
2                                     %    3"

Over the given interval draw the graphs of f and g in the same coordinate system. For the
graphs find to the nearest tenth the values of x for which f(x) = g(x)

&    '#
3. f(x) = sin x                                4. f(x) = cos \$ x + !
%    4"

&     3' #
g(x) = 2 cos x, 0 ! x < 2 !                     g(x) = sin \$ x +    !,
%      4 "
#! " x " !

Find the intersection of the set {t: 0 ! t < 2 ! } and the given set.

& 3#
5. arccos \$   !
\$ 2 !                                        6. arc csc 0
%   "

Name the least nonnegative member of each set.

&    '#
SAMPLE: arctan \$ cot !
%    5"

Solution:

)      &) ) #      3)          &    )#         &     3) #
cot     = tan\$ ( ! = tan    ;' arctan\$ cot ! = arctan\$ tan    !
5      %2 5"       10          %    5"         %     10 "

3!
The least nonnegative member of this set is      , Answer.
10

&    '#
7. arccos \$ cot !                                      8. arcsin[sin(-1)]
%    6"
Express in terms of x and y

9. cos (Arcsin 2x), x ! 0

Prove the identity.

4         1 !
10. Arctan      ! Arctan =
3         7 4

Solve the following open sentences for 0 ! x < 2 ! . Give approximations of values of x
to the nearest tenth and of ! to the nearest degree.

11. 2 sin ! sec ! = sec !                             12. 4 sin 4 x + 3 sin² x – 1 = 0

State the general solution of each of the following equations over the set of (a) real
numbers; (b) angles

13. sin x = cos x                            14. 4 sin ² x = 3 tan² x – 1

15. tan² x = 9 cos² x + sin² x

Solve over the real numbers for which " ! < x < ! .

16. sin x ! cos x

Solve the following equations for 0 ! x < 2 ! . Give approximations of values of x to the
nearest tenth and of ! to the nearest degree.

!
17. cos 2 ! (3 – 4 sin² ! ) = 0              18. tan ! = 3 tan
2
19. 3 sin 3 ! + 4 cos 3 ! = 5

Give the general solution of each equation. If the equation is an identity, state this fact
and prove it.

20. cos 4 3! " sin 4 3! = cos 6!

21. 1 – cos 6 5 x = sin 2 5 x(1 + cos 2 5 x + cos 4 5 x)

22. sin 2 ! tan² 2 ! - tan 2 ! = sin 2 !

1
23.     (2 ! sin 2 x ) = cos x
2
x   x          x
24. sin tan = 1 ! cos
2   4          2

cos(2 x + 1) ' cos 2 x     &        1          #
25.                          = 2 \$cos( x + ) + cos x !
1                %        2          "
cos( x + ) ' cos x
2

Express in terms of sine and cosine functions only, and simplify.

& cos ( ' sec (                   #& tan ( ' sin (   #
26. \$
\$               + cos 2 ( tan 2 ( !\$
!\$                 !
!
%     sec (                       "%     tan (       "

13             84
Find the value of the following expressions if sin ! =          and cos ! =
85             85

& sin ' sec ' + tan ' #
27. cos ' \$                     !
%     sec ' tan '     "

cot ! + cos !
28.
sec ! + tan !

Express as the cosine of the difference of the two angles, and evaluate.

29. cos 255 º

Verify as identities.

1
30. cos(150 º - ! ) = "
2
( 3 cos ! " sin ! )

Prove that the following are identities.

31. cos(180 º " ! ) = " cos !

Simplify:

32. tan(90 º - ! ) tan (180 º - ! ) sec ! + csc ! sin(90 º - ! )csc (90 º " ! )
Prove the Following Identities.

tan A ! sin A    tan A sin A
33.                 =
tan A sin A    tan A + sin A

34. If θ is a third-quadrant angleθwith sinθ=         and   a first quadrant angle with

sec          , find: a)                          b) cos ( - ! )

35. Evaluate                     for                              )

36. Prove: a)                                                     b)

37. Write as a sum or difference:

38. Verify the following identities:
a)

b)

39. Prove:
a.                                b.

c.                                    d.

e.

f.

g.                                          h.

i.                                         j.

k.                                               l.
m.

n.

o.   p.

q.

r.

s.

t.

```
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