Mathematics 5 SN TRIGONOMETRY - IDENTITIES Peter and Lucy are studying the repeated up and down movement of a piston in a gasoline engine. Peter claims that the piston moves according to the rule : t t t t t t d cos sin cos sin cos sin 2 2 2 Lucy believes that the piston moves according to the rule : d = sec t Prove that Peter and Lucy are both correct by showing that t t t t t t t sec cos sin cos sin cos sin 2 2 2 Note : t = time in seconds and d = distance in metres. Show your work. Work t t t t t t t sec cos sin cos sin cos sin 2 2 2 1
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Mathematics 5 SN TRIGONOMETRY - IDENTITIES...tan 1 cos 2 2 2 . Show your work. Work 9 . Prove the following identity : ... sec2 2cot2 1 = cot cot 1 cot cos 1 2 2 2 1 cot sin cos cos
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Mathematics 5 SN
TRIGONOMETRY - IDENTITIES
Peter and Lucy are studying the repeated up and down movement of a piston in a gasoline engine.
Peter claims that the piston moves according to the rule :
t
tt
t
ttd
cos
sincos
sin
cossin2 22
Lucy believes that the piston moves according to the rule :
d = sec t
Prove that Peter and Lucy are both correct by showing that
tt
tt
t
ttsec
cos
sincos
sin
cossin2 22
Note : t = time in seconds and d = distance in metres.
Show your work.
Work
tt
tt
t
ttsec
cos
sincos
sin
cossin2 22
1
Prove that,
xx
x
x
xx 2
2
2
2
22
tancos
cos1
1sec
1tansin
Show your work.
2
Work
x
x
x
x
xx 2
2
2
2
22
tancos
cos1
1sec
1tansin
Prove the following identity :
θsec2θsin1
θcos
θcos
θsin1
.
Show your work.
Work
θsec2θsin1
θcos
θcos
θsin1
3
Prove the following identity :
(1 + cot2)(1 cos2
) = 1.
Show your work.
Work
(1 + cot2)(1 cos2
) = 1
4
Prove the following identity :
sec2 cot2
1 = cot2.
Show your work.
Work
sec2 cot2
1 = cot2
5
Prove the following identity :
θsinθtan
θcosθsec
.
Show your work.
Work
θsinθtan
θcosθsec
6
Prove the following identity :
tan2 sin2
= sin2 tan2
.
Show your work.
Work
tan2 sin2
= sin2 tan2
.
7
Prove the following identity :
θcosecθsinθtan
θsec1
.
Show your work.
Work
θcosecθsinθtan
θsec1
8
Prove the following identity :
θcosθtan
θcos1 2
2
2
.
Show your work.
Work
θcosθtan
θcos1 2
2
2
9
Prove the following identity :
sec cos (sec 1) = cos .
Show your work.
Work
sec cos (sec 1) = cos .
10
Which expression is equivalent to (sec2t 1)(cosec2t 1)?
A)
0
C)
sin t
B)
1
D)
cos t
Which expression is equivalent to (1 sin2t)(1 cos2t)?
A)
0
C)
2sin2t cos2t
B)
1
D)
sin2t cos2t
If tan = ,3
5 what is the value of the following functions :
a) sin
b) cos
c) sec
d) cosec
e) cot
11
12
13
a) _________________________
b) _________________________
c) _________________________
d) _________________________
e) _________________________
Which expression is equivalent to sin2t sec2t sec2t?