D.E.V part 1

It’s been 3 years since the day Zack and Sally falling in love with each other.

Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise.

Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise.

Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise.

What should I do to make

her feel happy ?????

He’s thinking of ...

He’s thinking of ...He’s thinking of ...

Romantic night at the 5 stars restaurant.

He’s thinking of ...

He’s thinking of ...

Romantic night at the 5 stars restaurant.

Or going shopping!!!

He’s thinking of ...

Romantic night at the 5 stars restaurant.

Or going shopping!!!

Or a romantic movie

He’s thinking of ...

Romantic night at the 5 stars restaurant.

Or going shopping!!!

Or a romantic movieMake their love symbols

He’s thinking of ...

Romantic night at the 5 stars restaurant.

Or going shopping!!!

Or a romantic movieMake their love symbols

Or walking together at night

The answer is….

Hey! How about the Ferris Wheel??? That’s something new and really romantic….

Hey! How about the Ferris Wheel??? That’s something new and really romantic….

Hey! How about the Ferris Wheel??? That’s something new and really romantic….

Hey! How about the Ferris Wheel??? That’s something new and really romantic….

Hey that’s a great idea.

He starts planning everything for that special day.He starts planning everything for that special day.

That day is coming…..

They’re really happy spending time together….

Sally, you should try on that Ferris Wheel…

Sally, you should try on that Ferris Wheel…

Where honey???

Do you see it??..

Whoa..!!..

She gets on the Ferris Wheel, Zack said he has something for her and she has to get on the wheel in other to see that.

• Zack had planned to make a surprise for Sally. He will tell her to get on the Ferris wheel first, then Sally has to answer 2 questions in an definite time to see the special thing that Zack had prepared for only her….

• First he had to calculate the time he will give her to answer his 2 questions.

The Ferris Wheel has the maximum height is 32m and it stand 2m above from the round. It takes 6 minutes to go from the bottom of the wheel to the top of it. From the height of 28m, Sally will have the best view to see Zack’s present.

How long will it take to bring Sally to the best view point?

Solving:

Solving:

First we have to draw the graph so that we can easily solve this problem.

m

min

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

m

min

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

15

15

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

From the information above, we can sketch the graph.

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

From the information above, we can sketch the graph.

Sinusoidal axis

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

From the information above, we can sketch the graph.

Sinusoidal axis

P = 12mins12

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

From the information above, we can sketch the graph.

Sinusoidal axis

P = 12mins12

Max point

(An Amplitude)

Solving:

First we have to draw the graph so that we can easily solve this problem.

Maximum height : 32m

32

Stain 2m above the ground

2

m

min

Go from bottom to the top : 6minshalf period = 6mins

6

(32-2)

2= 15

[ the distance from the max and min points to the Sinusoidal Axis.]

17

From the information above, we can sketch the graph.

Sinusoidal axis

P = 12mins12

Max point

Min point

(An Amplitude)

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

A =

B =

C =

D =

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

A =

B =

C =

D =

15

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

15A =

B =

C =

D =

2πP

= 2π12

= π6

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

A =

B =

C =

D =

15

3 [the graph shifts to the right]

2πP

= 2π12

= π6

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

A =

B =

C =

D =

15

2πP

= 2π12

= π6

3 [the graph shifts to the right]

+17

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

A =

B =

C =

D =

152πP =

2π12 =

π6

+17

f(x) = A sin B (x – C) + D

A =

B =

C =

D =

2πP =

2π12 =

π6

15A =

B =

C =

D =

2πP =

2π12 =

π6

3 [the graph shifts to the right]

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

f(x) = A sin B (x – C) + D

15A =

B =

C =

D =

2πP =

2π12 =

π6

3 [the graph shifts to the right]

+17

π6

15 sin +17(x – 3)

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

f(x) = A sin B (x – C) + D

Sub the value f(x) = 28m we get:

15A =

B =

C =

D =

2πP =

2π12 =

π6

+17

3 [the graph shifts to the right]

π6

15 sin +17(x – 3)

32

2

m

min6

17

12

Sine Function: f(x) = A sin B (x – C) + D

3 [the graph shifts to the right]

A =

B =

C =

D =

15

2πP

= 2π12

= π6

+17

f(x) = A sin B (x – C) + D

Sub the value f(x) = 28m we get:

28 =

π6

15 sin +17(x – 3)

π6

15 sin +17(x – 3)

We know that:

We know that:

28 = π6

15 sin +17(x – 3)

We know that:

We know that:

sin π666

=

666666

1

2

28 = π6

15 sin +17(x – 3)

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = 15 x (x - 3) +1712

28 = π6

15 sin +17(x – 3)

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

28 = π6

15 sin +17(x – 3)

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = π6

15 sin +17(x – 3)

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

11 = 15x - 452

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = π6

15 sin +17(x – 3)

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

11 = 15x - 452

22 = 15x – 45

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = π6

15 sin +17(x – 3)

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

11 = 15x - 452

22 = 15x – 45

67 = 15x

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = π6

15 sin +17(x – 3)

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

11 = 15x - 452

22 = 15x – 45

67 = 15xx = 4.47 mins

We know that:

We know that:

sin π666

=

666666

1

2sin π

666=

666666

1

2sin π

666=

666666

1

2

28 = π6

15 sin +17(x – 3)

= 11 152

(x - 3)

28 = 15 x (x - 3) +1712

11 = 15x - 452

22 = 15x – 45

67 = 15xx = 4.47 mins ~ 4.5mins

We could also solve this with cosine function.

Therefore, Sally has about 4.5mins.

Sally has 3 boxes to choose that Zack had prepared before…

She picks the heart box first, and see a question.

If sinx = 712

If sinx = 712

Cosy = 12

If sinx = 712

Cosy = 12

Find cos2x + Sin2y

If sinx = 712

Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

If sinx = 712

Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2 = 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

Cos x2 -

Sin x2

Sin x2

Cos x2

= 1= 1= 1

712

= 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

= 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

= 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

= 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22 2Cos y

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22

Sin y22 = 1 -

2Cos y

2Cos y

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22

Sin y22 = 1 -

2Cos y

2Cos y

Sin y22 = 1 - 12

( (2

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22

Sin y22 = 1 -

2Cos y

2Cos y

Sin y22 = 1 - 12

( (2

Sin y22 = 1 - 14

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22

Sin y22 = 1 -

2Cos y

2Cos y

Sin y22 = 1 - 12

( (2

Sin y22 = 1 - 14

Sin y22 34=

If sinx = Cosy = 12

Find cos2x + Sin2y

From sinx, we could find cosx

Sin x2 + Cos x2

= 1

Cos x2 -

Sin x2

Sin x2

Cos x2

Cos x2

= 1= 1= 1

-

712

712

( (2

Cos x2 = 1 - 49144

Cos x2 = 95144

Cosx = √95144

= 1

With the same method, we could find siny

Sin y2 +2 2Cos y = 1Sin y22

Sin y22 = 1 -

2Cos y

2Cos y

Sin y22 = 1 - 12

( (2

Sin y22 = 1 - 14

Sin y22 34=

Siny = √32

49144

cos2x + sin2y =

Cos2x = Cos(x + x) Sin2y = sin(y + y)

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx

Sin2y = sin(y + y) = sinycosy + cosysiny

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32=

√32

√32

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y =

= √32

√32

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y = 46

144 +

= √32

√32

√32

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y = 46

144 +

= √32

√32

√32

=46

144 +72√3

144

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y = 46

144 +

= √32

√32

√32

=46

144 +72√3

144

46 + 72√3=144

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y = 46

144 +

= √32

√32

√32

=46

144 +72√3

144

46 + 72√3=144

2(23 + 36√3)144

=

cos2x + sin2y =

Cos2x = Cos(x + x)

= cosxcosx - sinxsinx= cos x – sin x2 2

= 95144

49144

-

= 46144

Sin2y = sin(y + y) = sinycosy + cosysiny

= 2sinycosy

= 2 x 12

√32

= √32

cos2x + sin2y = 46

144 +

= √32

√32

√32

=46

144 +72√3

144

46 + 72√3=144

2(23 + 36√3)144

=

=23 + 36√3

72

OMG!!! She has done with the first question, and starts to move on with the next one in the Noel box.

The next question is going to be harder than the first one…

The next question is going to be harder than the first one…

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

How can I solve this one??Uhm… let see…

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2

sinxcosx1 +sinxcosx1 +sinxcosx1 +

☺ sinxcosx1 + =

cosx + sinxcosx

cosx + sinxcosx

cosx + sinxcosx

cosx + sinxcosx

cosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

= cosx(cos x2 sin x2- )cosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

= cosx(cos x2 sin x2- )cosx + sinx

=cosx (cosx + sinx) (cosx – sinx)

cosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

= cosx(cos x2 sin x2- )cosx + sinx

=cosx (cosx + sinx) (cosx – sinx)

cosx + sinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

= cosx(cos x2 sin x2- )cosx + sinx

=cosx (cosx + sinx) (cosx – sinx)

cosx + sinx

= cosx (cosx – sinx)

● cotx =cosxsinx

cos2x1 + tanx

cotx 1 sin2x2

- 1 = + sin x2 -

●

cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x

=

= cos x2 -

sin x2

1 + tanxcos2x

1 + tanxcos2x

1 +sinxcosx

sin x2=

cos x2 sin x2-

cosxcosx + sinx

=

sin x2-cos x2 sin x2-

cos x2 sin x2-

1x

cosxcosx + sinx

= cosx(cos x2 sin x2- )cosx + sinx

=cosx (cosx + sinx) (cosx – sinx)

cosx + sinx

= cosx (cosx – sinx)

cos2x1 + tanx

cotx - 1 = cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x= 1 sin2x

2+ sin x2 -sin x2

cosxsinx

- 11 sin2x2+ sin x2 -sin x2

= cosx (cosx – sinx)

cos2x1 + tanx

cotx - 1 = cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x= 1 sin2x

2+ sin x2 -sin x2

cosxsinx

- 11 sin2x2+ sin x2 -sin x2

= cosx (cosx – sinx)

cosxsinx

- 1 = cos x2 - sinxcosx sinxcosx + sin x2 - 12

sin x2 12

- sinxcosx

cos2x1 + tanx

cotx - 1 = cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x= 1 sin2x

2+ sin x2 -sin x2

cosxsinx

- 11 sin2x2+ sin x2 -sin x2

= cosx (cosx – sinx)

cosxsinx

- 1 = cos x2 - sinxcosx + sin x2 -sin x2 sinxcosx 12

12

- sinxcosx

cosxsinx

- 1 = cos x2 sin x2+ - sinxcosx sinxcosx 12

12

- sinxcosx-

cos2x1 + tanx

cotx - 1 = cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x= 1 sin2x

2+ sin x2 -sin x2

cosxsinx

- 11 sin2x2+ sin x2 -sin x2

= cosx (cosx – sinx)

cosxsinx

- 1 = cos x2 - sinxcosx + sin x2 -sin x2 sinxcosx 12

12

- sinxcosx

cosxsinx

- 1 = cos x2 sin x2+ - sinxcosx sinxcosx 12

12

- sinxcosx-

cosxsinx

- 1 = 1 - 2sinxcosx

cos2x1 + tanx

cotx - 1 = cos2x1 + tanxcos2x

1 + tanxcos2x

1 + tanxcos2x= 1 sin2x

2+ sin x2 -sin x2

cosxsinx

- 11 sin2x2+ sin x2 -sin x2

= cosx (cosx – sinx)

cosxsinx

- 1 = cos x2 - sinxcosx + sin x2 -sin x2 sinxcosx 12

12

- sinxcosx

cosxsinx

- 1 = cos x2 sin x2+ - sinxcosx sinxcosx 12

12

- sinxcosx-

cosxsinx

- 1 = 1 - 2sinxcosxcosxsinx

- 1 = 1 - 2sinxcosxcosxsinx

- 1 = 1 -

cosxsinx

- 1 = 1 - sin2x

cosxsinx

- 1 = 1 - sin2x

cosxsinx

- 1 = 1 -

cosxsinx

- sinx

1 - sin2x

1 - sin2x=

cosxsinx

- 1 =

cosxsinx

- sinx

1 - sin2x

1 - sin2x=

cosx sinx- = sinx (1 - sin2x)

cosxsinx

- 1 =

cosxsinx

- sinx

1 - sin2x

1 - sin2x=

cosx sinx- = sinx (1 - sin2x)

cosx sinx- = sinx (cosx - sinx)2

cosxsinx

- 1 =

cosxsinx

- sinx

1 - sin2x

1 - sin2x=

cosx sinx- = sinx (1 - sin2x)

cosx sinx- = sinx (cosx - sinx)2

cosx sinx- sinx (cosx - sinx)2- = 0

cosxsinx

- 1 =

cosxsinx

- sinx

1 - sin2x

1 - sin2x=

cosx sinx- = sinx (1 - sin2x)

cosx sinx- = sinx (cosx - sinx)2

cosx sinx- sinx (cosx - sinx)2- = 0

(cosx - sinx) [1 – sinx(cosx – sinx)] = 0= 0=[1 – sinx(cosx – sinx)] 0=(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

Divide both sides by cos x2

X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1 –cos x2

tanx + tan x2

= 0=

0=

X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1cos x2

= 0=

– tanx + tan x2 0=tan x2

tan x2 + 1 – tanx + tan x2 0=tan x2

X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1cos x2

= 0=

– tanx + tan x2 0=tan x2

tan x2 + 1 – tanx + tan x2 0=tan x2

2tan x2 – tanx + 1 0=X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1cos x2

= 0=

– tanx + tan x2 0=tan x2

tan x2 + 1 – tanx + tan x2 0=tan x2

2tan x2 – tanx + 1 0=

= (-1) 2 - 4(2)(1)

X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1cos x2

= 0=

– tanx + tan x2 0=tan x2

tan x2 + 1 – tanx + tan x2 0=tan x2

2tan x2 – tanx + 1 0=

= (-1) 2 - 4(2)(1)

= 1 – 8 = -7

X = π4

5π4

X =

(cosx - sinx) [1 – sinx(cosx – sinx)] 0=

(cosx - sinx) = =0 0[1 – sinx(cosx – sinx)]

cosx = sinx1 – sinxcosx + sin x2 = 0

tanx = 1

(b/c given tanx = 1)

1cos x2

= 0=

– tanx + tan x2 0=tan x2

tan x2 + 1 – tanx + tan x2 0=tan x2

2tan x2 – tanx + 1 0=

= (-1) 2 - 4(2)(1)

= 1 – 8 = -7

< 0 [ O ]

X = π4

5π4

X =

X = π4

5π4

X =

X = π4

5π4

X =

● ●●

X = 5π4

_+ kπ

(K ͼ I)

● ●●

X = π4

_+ kπ

(K ͼ I)

4 mins 28 secs ….

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Made by TN – HN