P2 Trigonometry Watch

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Mathsman2
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#1
Report Thread starter 14 years ago
#1
1) sin (x+15) = 3cos (x-15).

2) 2sinxcosx = 1 - 2sin^2 x.

3) tan2x + tanx = 0.

4) 2 + cosxsinx = 8sin^2 x.

5) sin2x = cosx.

6) sec^2 x = 4tanx.

Answers:

1) 85.9, 265.9
2) 22.5, 112.5, 202.5, 292.5
3) 0, 60, 120, 180, 240, 300
4) 33.7, 153.4, 213.7, 333.4
5) 30, 90, 150, 270
6) 15, 75, 195, 255

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El Stevo
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#2
Report 14 years ago
#2
my i guide you to your previous thread... http://www.thestudentroom.co.uk/t83719.html
you use the same stuff in pretty much the same way...
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Nima
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#3
Report 14 years ago
#3
(Original post by Mathsman2)
Solve for 0 <= x <= 360:
1) sin (x+15) = 3cos (x-15).
2) 2sinxcosx = 1 - 2sin^2 x.
3) tan2x + tanx = 0.
4) 2 + cosxsinx = 8sin^2 x.
5) sin2x = cosx.
6) sec^2 x = 4tanx.
1.) sinxcos15 + cosxsin15 = 3cosxcos15 + 3sinxsin15
---> sinxcos15 + cosxsin15 - 3cosxcos15 - 3sinxsin15 = 0
---> sinx(cos15 - 3sin15) + cosx(sin15 - 3cos15) = 0
---> sinx(cos15 - 3sin15) = -cosx(sin15 - 3cos15)
---> tanx(cos15 - 3sin15) = -(sin15 - 3cos15)
---> tanx(cos15 - 3sin15) = 3cos15 - sin15
---> tanx = (3cos15 - sin15)/(cos15 - 3sin15)
---> tanx = 13.93
---> x = tan^-1(13.93) = 85.9 Deg
---> x = 180 + 85.9 = 265.9 Deg

Solutions: x = 85.9, 265.9 Deg

2.) 2sinxcosx = 1 - 2sin^2x
---> sin2x = cos2x
---> tan2x = 1
---> For interval 0 <= 2x <= 720:
---> 2x = 45 Deg
---> 2x = 225 Deg
---> 2x = 405 Deg
---> 2x = 585 Deg

Solutions: x = 22.5, 112.5, 202.5, 292.5 Deg

3.) tan2x + tanx = 0
---> sin2x/cos2x = -sinx/cosx
---> (2sinxcosx)/(1 - 2sin^2x) = -sinx/cosx
---> 2sinxcos^2x = 2sin^3x - sinx
---> 2sinx(1 - sin^2x) - 2sin^3x + sinx = 0
---> 2sinx - 2sin^3x - 2sin^3x + sinx = 0
---> 3sinx - 4sin^3x = 0
---> sinx(3 - 4sin^2x) = 0
---> sinx = 0, sin^2x = 3/4 ---> sinx = (+/-) Sqrt3/2
---> sinx = 0: x = 0, 180, 360 Deg
---> sinx = Sqrt3/2: x = 60, 120 Deg
---> sinx = -Sqrt3/2: x = 240, 300 Deg

Solutions: x = 0, 60, 120, 180, 240, 300, 360 Deg

5) sin2x = cosx
---> 2sinxcosx - cosx = 0
---> cosx(2sinx - 1) = 0
---> cosx = 0 Or sinx = 1/2
---> cosx = 0: x = 90, 270
---> sinx = 1/2: x = 30, 150

Solutions: x = 30, 90, 150, 270 Deg

6.) sec^2 x = 4tanx
---> 1 + tan^2x = 4tanx
---> tan^2x - 4tanx + 1 = 0
---> tanx = {4(+/-)Sqrt[16 - 4]}/2
= [4(+/-)2Sqrt3]/2
= 2(+/-)Sqrt3
---> tanx = 2 + Sqrt3: x = 75, 255 Deg
---> tanx = 2 - Sqrt3: x = 15, 195 Deg

Solutions: x = 15, 75, 195, 255 Deg

Question 4 is bloody difficult! You sure you wrote it properly?! mmm...
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Mathsman2
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#4
Report Thread starter 14 years ago
#4
(Original post by Nima)
1.) sinxcos15 + cosxsin15 = 3cosxcos15 + 3sinxsin15
---> sinxcos15 + cosxsin15 - 3cosxcos15 - 3sinxsin15 = 0
---> sinx(cos15 - 3sin15) + cosx(sin15 - 3cos15) = 0
---> sinx(cos15 - 3sin15) = -cosx(sin15 - 3cos15)
---> tanx(cos15 - 3sin15) = -(sin15 - 3cos15)
---> tanx(cos15 - 3sin15) = 3cos15 - sin15
---> tanx = (3cos15 - sin15)/(cos15 - 3sin15)
---> tanx = 13.93
---> x = tan^-1(13.93) = 85.9 Deg
---> x = 180 + 85.9 = 265.9 Deg

Solutions: x = 85.9, 265.9 Deg

2.) 2sinxcosx = 1 - 2sin^2x
---> sin2x = cos2x
---> tan2x = 1
---> For interval 0 <= 2x <= 720:
---> 2x = 45 Deg
---> 2x = 225 Deg
---> 2x = 405 Deg
---> 2x = 585 Deg

Solutions: x = 22.5, 112.5, 202.5, 292.5 Deg

3.) tan2x + tanx = 0
---> sin2x/cos2x = -sinx/cosx
---> (2sinxcosx)/(1 - 2sin^2x) = -sinx/cosx
---> 2sinxcos^2x = 2sin^3x - sinx
---> 2sinx(1 - sin^2x) - 2sin^3x + sinx = 0
---> 2sinx - 2sin^3x - 2sin^3x + sinx = 0
---> 3sinx - 4sin^3x = 0
---> sinx(3 - 4sin^2x) = 0
---> sinx = 0, sin^2x = 3/4 ---> sinx = (+/-) Sqrt3/2
---> sinx = 0: x = 0, 180, 360 Deg
---> sinx = Sqrt3/2: x = 60, 120 Deg
---> sinx = -Sqrt3/2: x = 240, 300 Deg

Solutions: x = 0, 60, 120, 180, 240, 300, 360 Deg

5) sin2x = cosx
---> 2sinxcosx - cosx = 0
---> cosx(2sinx - 1) = 0
---> cosx = 0 Or sinx = 1/2
---> cosx = 0: x = 90, 270
---> sinx = 1/2: x = 30, 150

Solutions: x = 30, 90, 150, 270 Deg

6.) sec^2 x = 4tanx
---> 1 + tan^2x = 4tanx
---> tan^2x - 4tanx + 1 = 0
---> tanx = {4(+/-)Sqrt[16 - 4]}/2
= [4(+/-)2Sqrt3]/2
= 2(+/-)Sqrt3
---> tanx = 2 + Sqrt3: x = 75, 255 Deg
---> tanx = 2 - Sqrt3: x = 15, 195 Deg

Solutions: x = 15, 75, 195, 255 Deg

Question 4 is bloody difficult! You sure you wrote it properly?! mmm...
Yep, I typed it in correctly Thanks a lot!
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