worksheet 6.2 trigonometric equations name:...maths quest maths b year 11 for queensland chapter 6:...

© 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

qq

980.0714.07.0

cossintan (b)

»

=

=qqq

9375.0

Maths Quest Maths B Year 11 for Queensland Chapter 6: Trigonometric equations WorkSHEET 6.2


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




Use Unit Circle:

sin 285 = −0.9659 (Must show Unit circle correctly)


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



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

worksheet 6.2 trigonometric equations name:...maths quest maths b year 11 for queensland chapter 6:...

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