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© John Wiley & Sons Australia, Ltd 2009 1
WorkSHEET 6.2 Trigonometric equations Name: ___________________________ 1 (a) State the Pythagorean identity.
(b) Write the Pythagorean identity with sin2x as the subject.
(c) Write the Pythagorean identity with cos2x as the subject.
(a) sin2x + cos2x = 1
(b) sin2x = 1 - cos2x
(c) cos2x = 1 - sin2x
3
2 If sin = 0.7, and 0o < < 90o, find, correct to three decimal places: (a) (b) *** TASK *** Make sure you can do this Mr Finney’s way as well and get an EXACTLY correct answer … J ** See Question 4.
4
3 Find all possible values of sin x if cos x = 0.25. ** as above … do this Mr Finney’s way ALSO! ** See Question 4.
sin2x = 1 - cos2x = 1 - (0.25)2 = 1 - 0.0625 = 0.9375 sin x = ± = 0.968 or -0.968
3
q q
qcos
qtan
714.051.0cos
51.049.01
)7.0(1
sin1cos (a)2
22
»=
=-=-=
-=
q
980.0714.07.0
cossintan (b)
»
=
=qqq
9375.0
Maths Quest Maths B Year 11 for Queensland Chapter 6: Trigonometric equations WorkSHEET 6.2
© John Wiley & Sons Australia, Ltd 2009 2
4 Find the exact value of sin x if cos x = and x
is in the fourth quadrant. *** Refer Q 2 & 3 notes ***
Using Pythagoras’ theorem, third side of triangle = .
From triangle, sin x = but x is in the
fourth quadrant, so sin x = - .
3
5 If 0o £ a £ 90o and sin a = , find the exact
value of : (a) cos a
(b) tan a (c) cos (180 + a)o
By Pythagoras’ theorem, third side of triangle = .
(a) cos a =
(b) tan a =
(c) cos (180 + a)o = -cos a
cos (180 + a)o = -
4
6 If cos 20 = 0.9397, manually determine the value of cos 160.
Use Unit Circle:
cos 160 = −0.9397 (Must show Unit circle correctly)
7 If sin 75 = 0.9659, manually determine the value of sin 255.
Use Unit Circle:
sin 255 = −0.9659 (Must show Unit circle correctly)
52
21
521
521
31
8
38
81
38
Maths Quest Maths B Year 11 for Queensland Chapter 6: Trigonometric equations WorkSHEET 6.2
© John Wiley & Sons Australia, Ltd 2009 3
8 If sin 75 = 0.9659, manually determine the value of sin 285.
Use Unit Circle:
sin 285 = −0.9659 (Must show Unit circle correctly)
9 If sin 75 = 0.9659, manually determine the value of sin 105.
Use Unit Circle:
sin 105 = 0.9659 (Must show Unit circle correctly)
10 If tan 25 = 0.4663, manually determine the value of tan 155.
Use Unit Circle:
tan 155 = −0.4663 (Must show Unit circle correctly)
11 If tan 25 = 0.4663, manually determine the value of tan 205.
Use Unit Circle:
tan 205 = 0.4663 (Must show Unit circle correctly)
12 Solve the equation 2 sin2x = sin x over the domain 0 £ x £ 2
2 sin2x = sin x 2 sin2x - sin x = 0 sin x (2 sin x - 1) = 0 sin x = 0 2 sin x - 1 = 0 x = 0, 2 2 sin x = 1
sin x =
x = ,
Solution is x = 0, , , p, 2 .
4
13 Solve the equation 2cos2x + cos x = 0 for the domain 0 £ x £ 2
2cos2x + cos x = 0 cos x (2 cos x + ) = 0 cos x = 0 2 cos x + = 0
x = , 2 cos x = -
cos x =
x = ,
Solution is x = , , , .
4
.p
,p p
21
6p
65p
6p
65p p
3.p
33
3
2p
23p 3
23
-
65p
67p
2p
65p
67p
23p
Maths Quest Maths B Year 11 for Queensland Chapter 6: Trigonometric equations WorkSHEET 6.2
© John Wiley & Sons Australia, Ltd 2009 4
14 Solve 2 sin2x = 3 sin x - 1 in the domain 0 £ x £ 2
2 sin2x = 3 sin x - 1 2 sin2x - 3 sin x + 1 = 0 (2 sin x - 1)(sin x -1) = 0 2 sin x - 1 = 0 sin x - 1 = 0 2 sin x = 1 sin x = 1
sin x = x =
x = ,
Solution is x = , , .
4
15 Solve 2sin2x = 2 - cos x in the domain 0 £ x £ 2
2sin2x = 2 - cos x 2(1 - cos2x) = 2 - cos x 2cos2x - cos x = 0 cos x (2cos x - ) = 0 cos x = 0 2 cos x - = 0
x = , 2 cos x =
cos x =
x = ,
Solution is x = , , , .
4
.p
21
2p
6p
65p
6p
2p
65p
3.p
33
33
3
2p
23p 3
23
6p
611p
6p
2p
23p
611p