Problem no.1Problem no.1

Trig IdentitiesTrig Identities

Can u help me to prove this trig identity?Can u help me to prove this trig identity?

In solving these problems, you have to work on the complicated side In solving these problems, you have to work on the complicated side first. But in this case both of them are complicated so we will solve first. But in this case both of them are complicated so we will solve them together.them together.

Firstly, you have to put a Firstly, you have to put a boundary boundary

between them or what we called between them or what we called our our

““Red Great Wall of China”.Red Great Wall of China”.Left sideLeft side::We recognized that sin2x is a We recognized that sin2x is a

double double angle identity and so as the angle identity and so as the

cos2xcos2xRight sideRight side::cot²x+1=csc²xcot²x+1=csc²x

Left side:Left side:I will write the long term for I will write the long term for

double double angle identities to make things angle identities to make things

clear clear for you.for you.

Right side:Right side:We’ll leave cotx the way it isWe’ll leave cotx the way it is(there’s a reason for that)(there’s a reason for that)Then, sec²x=1/cos²xThen, sec²x=1/cos²xcsc²x=1/sin²xcsc²x=1/sin²x

Left side:Left side:(Numerator)(Numerator) We can add We can add

sinxcosx+cosxsinx since they are sinxcosx+cosxsinx since they are same terms..same terms..

(Denominator)(Denominator)cosx · cosx= cos²x cosx · cosx= cos²x sinx · sinx= sin²xsinx · sinx= sin²x

So it will equal to 1+cos²x-sin²xSo it will equal to 1+cos²x-sin²xWe could recognize that We could recognize that 1-sin²x=cos²x1-sin²x=cos²x

Right side:Right side:We can already change cotx=cosx/sinxWe can already change cotx=cosx/sinxThe equation is divided by 1/sin²x, we The equation is divided by 1/sin²x, we

can take the reciprocal of it and can take the reciprocal of it and multiply it to the equation.multiply it to the equation.

Left side:Left side:2 is reduced and cos as well. 2 is reduced and cos as well.

We’ll We’ll get sinx/cosxget sinx/cosx

Right side:Right side:so the equation is reduced to so the equation is reduced to sinx/cosxsinx/cosx

Eventhough they are already the Eventhough they are already the same we can still reduce it… same we can still reduce it…

Sinx/cosx = tanxSinx/cosx = tanx

Thanks for your help..hehe..Thanks for your help..hehe..

Problem no. 2Problem no. 2

PermutationsPermutations

I’m J.r. Manleigh and I am the manager of the new band called I’m J.r. Manleigh and I am the manager of the new band called “Private”. I have here the list of the possible provinces that we can “Private”. I have here the list of the possible provinces that we can have the concert.have the concert.

Luzon(red)Luzon(red) Visayas(yellowVisayas(yellow))

Mindanao(greeMindanao(green)n)

LagunaLaguna CebuCebu Surigao Del Surigao Del SurSur

CaviteCavite BoholBohol Surigao Del Surigao Del NorteNorte

La UnionLa Union RomblonRomblon SuluSuluBatangasBatangas LeyteLeyte BukidnonBukidnonBataanBataanPangasinanPangasinan

If the band plays 6 provinces in total, how many different concert If the band plays 6 provinces in total, how many different concert schedules can be created if there is no restrictions in which they schedules can be created if there is no restrictions in which they

play?play?Order is important thing in a touring schedule, so use permutations. Order is important thing in a touring schedule, so use permutations. Out of 12 possible provinces, the band wants to create a schedule Out of 12 possible provinces, the band wants to create a schedule

with with only 6. only 6.

Pick FormulaPick Formula:: nnPPr r = = nn!! ((n n - - rr)!)! nPr is read as “n Pick r”nPr is read as “n Pick r” where where nn is the number of objects to pick from is the number of objects to pick from rr is the no. of objects to be arranged is the no. of objects to be arranged

nnPPrr = = nn!! ((nn--rr)!)!

1414PP66 = = 1414!! ((1414--66)!)! 1414!! 8!8! = = 2,162,1602,162,160

or using your calculator…or using your calculator…

1414PP66 = = 2,162,1602,162,160 different concert schedules different concert schedules

If the band the band must play all provinces in Luzon, 3 in Visayas If the band the band must play all provinces in Luzon, 3 in Visayas and 2 in Mindanao how many different schedules can be created?and 2 in Mindanao how many different schedules can be created?

The 6 provinces in The 6 provinces in LuzonLuzon can be ordered in can be ordered in 66PP66 ways. ways. The 3 provinces in The 3 provinces in Visayas Visayas can be ordered in can be ordered in 33PP44 ways. ways.The 2 provinces in The 2 provinces in MindanaoMindanao can be ordered in can be ordered in 22PP44 ways. ways.

6P6P66 · 4P · 4P33 · 4P · 4P22 = 207,360 different = 207,360 different schedulesschedules

If the band decides not to go to If the band decides not to go to BataanBataan and and Sulu Sulu (because many (because many rumors about those provinces that they are anti-Private people). How rumors about those provinces that they are anti-Private people). How many different schedules can be created if the band plays all many different schedules can be created if the band plays all provinces in provinces in LuzonLuzon and and MindanaoMindanao, and only 3 for , and only 3 for VisayasVisayas??

We can already ignore We can already ignore BataanBataan and and SuluSulu from the selection pool. from the selection pool.The first 5 cities in The first 5 cities in LuzonLuzon can now be ordered in 5P5 ways. can now be ordered in 5P5 ways.The 3 cities in The 3 cities in Visayas Visayas can be ordered in 4P3 ways.can be ordered in 4P3 ways.The 3 cities in The 3 cities in MindanaoMindanao can be ordered in 3P3 ways. can be ordered in 3P3 ways.

So…So…

5P5 · 4P3 · 3P3 = 17280 ways that the 5P5 · 4P3 · 3P3 = 17280 ways that the concert schedule can be createdconcert schedule can be created

Problem no. 3Problem no. 3LogarithmsLogarithms

Rio really likes the model of Honda Civic 2007 which will cost you a Rio really likes the model of Honda Civic 2007 which will cost you a brand new one $24, 950 but cannot afford to buy it now. She’s brand new one $24, 950 but cannot afford to buy it now. She’s planning to save up some money first then buy it. After 3 years, she planning to save up some money first then buy it. After 3 years, she will have enough money to buy a used car Honda Civic 2007. But she will have enough money to buy a used car Honda Civic 2007. But she has 3 dealers to choose from if she wants to get a low priced car. has 3 dealers to choose from if she wants to get a low priced car.

Andrei who offers a Civic that Andrei who offers a Civic that depreciates 0.02% daily for 3 depreciates 0.02% daily for 3 years.years.

Danny who offers a Civic that Danny who offers a Civic that depreciates 0.5% monthly for 3 depreciates 0.5% monthly for 3 years.years.

Ronnie who offers a Civic that Ronnie who offers a Civic that depreciates for 0.15% weekly depreciates for 0.15% weekly for 3 years.for 3 years.

Who will be the best to go to if Who will be the best to go to if she wants the cheapest one? she wants the cheapest one?

Andrei who offers a Civic that depreciates 0.02% daily for 3 Andrei who offers a Civic that depreciates 0.02% daily for 3 years.years.

AAo = o = 24, 95024, 950

model = 0.02% model = 0.02% → → 0.0002 (percent to decimal)0.0002 (percent to decimal)t = 365 x 3(because 365 days in a year and it is for 3 years)t = 365 x 3(because 365 days in a year and it is for 3 years)A = ?A = ?p = 1p = 1

A = 20, 042.4379A = 20, 042.4379 oror $20, 042.44$20, 042.44

Danny who offers a Civic that depreciates 0.5% monthly for 3 Danny who offers a Civic that depreciates 0.5% monthly for 3 years.years.

AAo = o = 24, 95024, 950model = 0.5% model = 0.5% → → 0.005 (percent to decimal)0.005 (percent to decimal)t = 12 x 3 (because 12 months in one year and it is for 3 years)t = 12 x 3 (because 12 months in one year and it is for 3 years)A = ?A = ?p = 1p = 1

A = 20, 830.5845A = 20, 830.5845 oror $20, 830.58$20, 830.58

Ronnie who offers a Civic that depreciates for 0.15% weekly for Ronnie who offers a Civic that depreciates for 0.15% weekly for 3 3

years.years.

AAo = o = 24, 95024, 950

model = 0.15% model = 0.15% → → 0.0015 (percent to decimal)0.0015 (percent to decimal)t = 52 x 3 (because 52 weeks in one year and it is for 3 years)t = 52 x 3 (because 52 weeks in one year and it is for 3 years)A = ?A = ?p = 1p = 1

A = 19, 741.0090A = 19, 741.0090 oror $19, 741.01$19, 741.01

We all know that Ronnie is the cheapest one We all know that Ronnie is the cheapest one So Riolyn will choose him.So Riolyn will choose him.

Problem no. 4Problem no. 4ConicsConics

What you see above is a tunnel in Winnipeg that has a length of 162 What you see above is a tunnel in Winnipeg that has a length of 162 m and a height of 56m and a height of 56 m.m.

If the arch is modeled as an ellipse with the standard form:If the arch is modeled as an ellipse with the standard form: (x - h)² + (y - k)² (x - h)² + (y - k)² = 1= 1

a² b²a² b²Determine the equation:Determine the equation:First, is to use the measures that was given to you.First, is to use the measures that was given to you.Major axis: 2a = 162 Major axis: 2a = 162 a= 81a= 81Semi-minor axis: b = 54 Semi-minor axis: b = 54 Centre = (0, 0)Centre = (0, 0)Therefore: Therefore: (x - h)² + (y - k)² = 1 (x - 0)²+ (y – 0)² = 1(x - h)² + (y - k)² = 1 (x - 0)²+ (y – 0)² = 1 a² b² (81)² (54)²a² b² (81)² (54)²

x² + y² = 1x² + y² = 1 6561 2916 6561 2916

If the arch is modeled as a parabola with the standard form: If the arch is modeled as a parabola with the standard form: y – k = a(x - h)²y – k = a(x - h)²

Determine the equation:Determine the equation:First state what you know: Vertex = (0 , 54)First state what you know: Vertex = (0 , 54) point in the graph = (81, 0)point in the graph = (81, 0)

Try to plug these values in the equation to find the “a” value.Try to plug these values in the equation to find the “a” value.y - k = a( x - h )²y - k = a( x - h )²0 - 54 = a(81 - 0) ²0 - 54 = a(81 - 0) ² -54 = a(81) ²-54 = a(81) ² -54 = 6561 a-54 = 6561 a 6561 65616561 6561

a = -54 a = -54 65616561y – 54 = -54 x²y – 54 = -54 x² 65616561

Determine the height of the tunnel at a point 40 m away from the Determine the height of the tunnel at a point 40 m away from the center using center using

the equations given above. If the actual height at this point is 40.2 m, the equations given above. If the actual height at this point is 40.2 m, determine which model makes the closest approximation.determine which model makes the closest approximation.

Ellipse: Parabola:Ellipse: Parabola: x² + y² = 1 y-54 = -54xx² + y² = 1 y-54 = -54x²² 6561 2916 65616561 2916 6561

1600 + y² = 1 y-54 = -54(40)1600 + y² = 1 y-54 = -54(40)²²

6561 2916 65616561 2916 6561

y² = 1 – 0.244 y-54 = -13.169y² = 1 – 0.244 y-54 = -13.169 2916 2916

y=40.83y=40.83 y² = 0.756y² = 0.756 29162916 √ √ y² = √ 2204.496y² = √ 2204.496 y =46.952y =46.952

Ellipse = 46.952Ellipse = 46.952Parabola = 40.83Parabola = 40.83

Which makes parabola the closest one to the actual Which makes parabola the closest one to the actual height.height.