11 x1 t08 07 asinx + bcosx = c (2012)
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Equations of the form asinx + bcosx = cMethod 1: Using the t results
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2
let tan2
t
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1 4 21tan5
59 47
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1 4 21tan5
59 47
59 472
119 33
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1 4 21tan5
59 47
59 472
119 33
180: Test
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1 4 21tan5
59 47
59 472
119 33
180: Test
3cos180 4sin1804 2
Equations of the form asinx + bcosx = cMethod 1: Using the t results
3600 eg (i) 3cos 4sin 2 2
2 21 23 4 2 0 1801 1 2
t tt t
let tan
2t
2 23 3 8 2 2t t t 25 8 1 0t t
8 8410
t
4 21 4 21tan or tan2 5 2 5
Q2 21 4tan5
6 39
173 212
346 42
Q1 4 21tan5
59 47
59 472
119 33
180: Test
3cos180 4sin1804 2
119 33 ,346 42
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
sin corresponds to 3, so 3 goes on the opposite
side
3
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
cos corresponds to 4, so 4 goes on the adjacent
side
4
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
4
by Pythagoras the hypotenuse is 5
5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
4
by Pythagoras the hypotenuse is 5
5
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
4
by Pythagoras the hypotenuse is 5
5
5sin 2
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
sincoscossin
3
4
by Pythagoras the hypotenuse is 5
5
5sin 2
The hypotenuse becomes the coefficient of the trig function
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
2sin5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
2sin5
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
2sin5
23 35
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
2sin5
23 35
36 52 23 35 ,156 25
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
2sin5
23 35
36 52 23 35 ,156 25
13 17 ,119 33
Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
3
4
5
5sin 2
3 45 cos sin 25 5
3tan
4
36 52 2sin5
Q1, Q2
2sin5
23 35
36 52 23 35 ,156 25 119 33 ,346 43
13 17 ,119 33
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
cos corresponds to 3, so 3 goes on the adjacent
side
3
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
cos corresponds to 4, so 4 goes on the adjacent
side
4
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
4
by Pythagoras the hypotenuse is 5
5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
4
by Pythagoras the hypotenuse is 5
5
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
4
by Pythagoras the hypotenuse is 5
5
5cos 2
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
4
by Pythagoras the hypotenuse is 5
5
5cos 2
The hypotenuse becomes the coefficient of the trig function
3 45 cos sin 25 5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
2cos5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Q1, Q4
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Q1, Q4
2cos5
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Q1, Q4
2cos5
66 25
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Q1, Q4
2cos5
66 25
53 8 66 25 ,293 35
Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600
3cos 4sin 2
cos cos sin sin
3
45
5cos 2
3 45 cos sin 25 5
4tan
3
53 8 2cos5
Q1, Q4
2cos5
66 25
53 8 66 25 ,293 35
119 33 ,346 43
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
cos sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
1
12 cos sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
1
12
2 cos x cos sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
1
12
2 cos x
tan 145
cos sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
1
12
2 cos x
tan 145
2 cos 45x
cos sinx x
(ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)
3sin3 cos3t t sin 3 cos cos3 sint t
3
12 2sin 3t
1tan3
30
2sin 3 30t
(iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)
sin cosx x cos cos sin sinx x
1
12
2 cos x
tan 145
2 cos 45x
cos sinx x
Exercise 2E;6, 7, 10bd, 11, 13, 14, 16ac, 20a, 23
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