Download - 11 x1 t08 05 t results (2012)
The t results
The t resultsLet tan
2t
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t ;2
tan If t
212tan
tt
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t ;2
tan If t
212tan
tt
21
2sintt
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t ;2
tan If t
212tan
tt
21
2sintt
2
2
11cos
tt
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t ;2
tan If t
212tan
tt
21
2sintt
2
2
11cos
tt
Note:If tan ;t 2
2tan 21
tt
The t resultsLet tan
2t
2
2 tan2tan
1 tan2
212tan
tt
t2
21 t
21 t
222 22 1h t t 2 2 44 1 2t t t
4 22 1t t
22 1t ;2
tan If t
212tan
tt
21
2sintt
2
2
11cos
tt
Note:If tan ;t 2
2tan 21
tt
If tan 2 ;t 22tan 4
1tt
1 cose.g. i Show that , where tansin 2
x xt tx
1 cose.g. i Show that , where tansin 2
x xt tx
1 cossin
xx
2
2
2
1112
1
tt
tt
is double , 2
so results can be used for sin ,cos
xx
t x x
1 cose.g. i Show that , where tansin 2
x xt tx
1 cossin
xx
2
2
2
1112
1
tt
tt
is double , 2
so results can be used for sin ,cos
xx
t x x
2 21 12
t tt
1 cose.g. i Show that , where tansin 2
x xt tx
1 cossin
xx
2
2
2
1112
1
tt
tt
is double , 2
so results can be used for sin ,cos
xx
t x x
2 21 12
t tt
222tt
t
22 tan 75ii Use tan to simplify
2 1 tan 75t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sin
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
Let tan ;2
t 1 sin cos
1 sin cos
2
2 2
2
2 2
2 111 1
2 111 1
t tt tt tt t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
Let tan ;2
t 1 sin cos
1 sin cos
2
2 2
2
2 2
2 111 1
2 111 1
t tt tt tt t
2 2
2 2
1 2 11 2 1
t t tt t t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
Let tan ;2
t 1 sin cos
1 sin cos
2
2 2
2
2 2
2 111 1
2 111 1
t tt tt tt t
2 2
2 2
1 2 11 2 1
t t tt t t
22 2
2 2t t
t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
Let tan ;2
t 1 sin cos
1 sin cos
2
2 2
2
2 2
2 111 1
2 111 1
t tt tt tt t
2 2
2 2
1 2 11 2 1
t t tt t t
22 2
2 2t t
t
2 12 1t t
t
22 tan 75ii Use tan to simplify
2 1 tan 75t
Let tan 75 ;t
i.e. 752
150
22 tan 75
1 tan 75
22
1tt
sinsin150
12
1 sin cosiii Prove tan1 sin cos 2
Let tan ;2
t 1 sin cos
1 sin cos
2
2 2
2
2 2
2 111 1
2 111 1
t tt tt tt t
2 2
2 2
1 2 11 2 1
t t tt t t
22 2
2 2t t
t
2 12 1t t
t
t tan2
2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,2
show that cosec cot cot2
t
2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,2
show that cosec cot cot2
t
cosec cot 2 21 1
2 2t tt t
2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,2
show that cosec cot cot2
t
cosec cot
t
t122
2 21 12 2
t tt t
2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,2
show that cosec cot cot2
t
cosec cot
t
t122
2cot
2 21 12 2
t tt t
2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,2
show that cosec cot cot2
t
cosec cot
t
t122
2cot
2 21 12 2
t tt t
Exercise 2B; 1def, 3ace, 4bcf, 5aceg, 6, 8bd, 10a
