Download Worksheet 6.2

Maths Quest Maths B Year 11 for Queensland
WorkSHEET 6.2
Chapter 6 Trigonometric equations WorkSHEET 6.2
1
Trigonometric equations
Name: ___________________________
1
(a)
State the Pythagorean identity.
(a)
sin2x + cos2x = 1
(b)
Write the Pythagorean identity with sin2x (b)
as the subject.
(c)
Write the Pythagorean identity with cos2x
as the subject.
sin2x = 1  cos2x
(c)
2
If sin  = 0.7, and 0o <  < 90o, find, correct to (a)
three decimal places:
(a) cos 
(b)
3
cos2x = 1  sin2x
cos 2   1  sin 2 
4
 1  (0.7) 2
 1  0.49
tan 
 0.51
cos   0.51
 0.714
(b)
3
Find all possible values of sin x if cos x = 0.25.
sin 
cos 
0 .7

0.714
 0.980
tan  
sin2x = 1  cos2x
= 1  (0.25)2
= 1 0.0625
= 0.9375
sin x =  0.9375
x = 0.968 or 0.968
3
Maths Quest Maths B Year 11 for Queensland
4
Find the exact value of sin x if cos x =
Chapter 6 Trigonometric equations WorkSHEET 6.2
2
3
2
and x
5
is in the fourth quadrant.
Third side of triangle =
21
21
From triangle sin x =
but x is in the fourth
5
21
quadrant, so sin x = 
.
5
5
Find ao if 0o <  < 90o and
(a) sin ao = cos 67o
(b)
6
(a)
sin ao = cos 67o
= sin (90 – 67)o
= sin 23o
ao = 23o
(b)
cos ao = sin 8o
= cos (90 – 8)o
= cos 82o
ao = 82o
cos ao = sin 8o
If 0o  a  90o and sin ao =
2
4
1
, find the exact
3
value of :
(a) cos ao
(b)
tan ao
(c)
sin (90 – a)o
Third side of triangle =
(d)
cos (180 + a)o
(a)
cos ao =
(b)
tan ao =
(c)
(d)
8
8
3
1
8
sin (90 – a)o = cos ao
8
=
3
o
cos (180 + a) = cos ao
8
=
3
Maths Quest Maths B Year 11 for Queensland
Chapter 6 Trigonometric equations WorkSHEET 6.2
3
7
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
1
sin x =
2
 5
x= ,
6 6

5
Solution: x = 0, , ,
, 2
6
6
4
8
Solve the equation 2cos2x +
the domain 0  x  2  .
2cos2x + 3 cos x = 0
cos x (2 cos x + 3 ) = 0
cos x = 0
2 cos x + 3 = 0
 3
x= ,
2 cos x = 
2 2
4
3 cos x = 0 for
cos x = 
x=
Solution: x =
9
Solve 2 sin2x = 3 sin x  1 in the domain
0  x  2 .
3
3
2
5 7
,
6
6
 5 7
3
,
,
or
2
6
2 6
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
 3
1
sin x =
x= ,
2
2 2
 5
x= ,
6 6
  5
3
Solution: x = , ,
or
2
6 2 6
4
Maths Quest Maths B Year 11 for Queensland
10
Solve 2sin2x = 2 
0  x  2 .
Chapter 6 Trigonometric equations WorkSHEET 6.2
3 cos x in the domain
2sin2x = 2 
3 cos x
2
2(1  cos x) = 2  3 cos x
2cos2x  3 cos x = 0
cos x (2cos x  3 ) = 0
cos x = 0
2 cos x  3 = 0
 3
x= ,
2 cos x = 3
2 2
3
cos x =
2
 11
x= ,
6
6
  3
11
Solution: x = , ,
or
6
6 2 2
4
4