Worksheet 3 Trigonometry by vcl99353

VIEWS: 33 PAGES: 5

									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.

								
To top