trigonometric.pdf

1

e-Learning

.

18 2555 09.00 15.00 .

.

2

1.

(x1, y1) (x2, y2 ) 2 22 1 2 1( ) ( )x x y y

(1,2) (5,-1 ) 2 2(5 1) ( 1 2) 16 9 5

(x1, y1) (x2, y2 ) 1 2 1 2,2 2

x x y y

2.

2.1

y = mx + c m, x ( cm

, 0)

y (0, c)

Ax + By + C = 0 AB

, x ( CA

,0)

y (0, CB

)

2.2 (x0,y0) m y y0 = m(x x0)

2.3 (x0,y0) (x1,y1) y y0 = 1 01 0

y y

x x

(x x0)

(x0,y0) (x1,y1) 1 01 0

y y

x x

3

2.4 0 () x

0 () x x y

2.5

-1

2.6

Ax + By + C = 0 (x0,y0) 0 02 2

| |Ax By C

A B

2.7

Ax + By + C = 0 Ax + By + D = 0

2 2

| |C D

A B

3. 3.1

r (h,k)

4

3.2 (h,k) r 2 2 2( ) ( )x h y k r 2 2 0x y Ax By C 2 2 4 0A B C A, B, C

,2 2

A B

2 21 42

A B C

4.

4.1 ()

5

4.2 (h,k) x 2( ) 4 ( )y k c x h

c > 0 c < 0 (h + c, k)

x = h c y = k |4c| 2 0y Ay Bx C

A, B, C B 0 (2 4

4

A C

B

,2

A )

4.3 (h,k) y 2( ) 4 ( )x h c y k

c > 0 c < 0 (h, k + c)

y = k c x = h |4c| 2 0x Ax By C

A, B, C B 0 (2

A ,

2 4

4

A C

B

)

6

5. 5.1

F1 , F2 A, B C, D

5.2 (h,k) x

2 2

2 2

( ) ( )1

x h y k

a b

, a > b

(h + a , k) (h a , k) (h + c , k) (h - c , k) 5.3 (h,k) y

2 2

2 2

( ) ( )1

y k x h

a b

, a > b

(h , k + a) (h, k a) (h , k + c) (h, k - c)

7

5.4 x y 2a 2b a2 = b2 + c2

22b

a

6.

6.1 F1 , F2 A, B C, D L1, L2

6.2 (h,k) x

F1 F2 A B

C

D

L2

L1

8

2 2

2 2

( ) ( )1

x h y k

a b

(h + a , k) (h a , k) (h + c , k) (h - c , k)

y = ( )bk x ha

6.3 (h,k) y

2 2

2 2

( ) ( )1

y k x h

a b

(h , k + a) (h, k a) (h , k + c) (h, k - c)

y = ( )ak x hb

6.4 x y 2a 2b c2 = a2 + b2

22b

a

9

1. f(x) = 2x2 4x 6

. x = -1 .

2. f(x) = -x^2 + x + 2 . f(x) 0 -1 x 2 . f . f 2 . f 2

10

3. x y = -4x2 5x + 6 x

. (- 23

, - 13

)

. (- 52

, - 32

)

. ( 14

, 67

)

. ( 12

, 32

)

4. x = 3 f(x) = -x2 + (k + 5)x + (k2 - 10) k f

5. y (a,b)

x (-1,0) (5,0) a

11

6. y = x2 2x 8 x A, B C

ABC

7. f(x) = -x2 + 4x 10 . f -6 . f . f 6

. f( 92

) < -6

8. P y = -x2 + 12x 38 O P O

9. (3,1) (3,6)

(a,6) x (b,0) b

12

10. y = x2 2x + 4 x 2 y 1

11. . x2 + y2 + 6x 4y = 23

21x + 20y + 168 = 0 . y2 + 16x 6y = 71 (-5,3)

(-1,3)

12. A(1,0) B(b,0) b > 1

L (-1,0) 43

b

13

13. x y + 2 = 0 x2 + y2 + 6x 4y + 4 = 0 A B (a,b)

y = 2 A B a + b

14.

() 4x2 25y2 + 24x 100y 164 = 0 4x2 + 25y2 + 24x + 100y + 36 = 0

() 4x2 + 25y2 + 24x + 100y + 36 = 0 y2 = 4y 4x + 12 = 0

15. 25x2 + 21y2 + 100x -42y 404 = 0 (-3,1 + 8 )

) 5y2 4x2 - 10 8 y 32x 25 = 0 ) 3y2 2x2 - 6 8 y 8x + 15 = 0 ) y2 4x2 - 2y 16x - 19 = 0 ) y2 7x2 - 2y 28x - 28 = 0

14

16. A(-3,1), B(1,5), C(8,3) D(2,-3) ABCD

) AB DC ) AB DC 10 2

) A C D 9 22

) B C D 9

2

17. ABC A(0,0) B(2,2) C(x,y) 2 AC BC ABC 4 C

) x y + 4 = 0 ) 4x + 3y 1 = 0 ) 2x y -3 = 0 ) x + y 5 = 0

18. A B 4x2 y2 24x + 6y + 11 = 0 AB x

) (x-3)2 = 4(y-2) ) (x-3)2 = 8(y-1) ) (x-2)2 = 4(y-2) ) (x-2)2 = 8(y-1)

15

19. 20. 21.

16

22. 23.

17

1. 1.1 (0,0) 1 1.2 () (1,0)

(x,y) > 0 () (1,0) < 0 () (1,0)

(x,y) x = cos y = sin

1.3 (x,y) { (,y) : y = sin } { (,y) : y = cos }

1.4

cos2 + sin2 = 1 -1 cos 1

-1 sin 1 1.5

1) sin(- ) = -sin 2) cos(- ) = cos

3) sin( - ) = sin 0 < < 2

4) cos( - ) = -cos 0 < < 2

18

5) sin( + ) = -sin 0 < < 2

6) cos( + ) = -cos 0 < < 2

7) sin(2 - ) = -sin 0 < < 2

8) cos(2 - ) = cos 0 < < 2

1.6

tan = sincos

sec = 1

cos

cosec = 1

sin

cot = cos

sin

1.7

180 =

19

2. A, B

1) cos2A + sin2B = 1 2) sin(A+B) = sin A cos B + cos A sin B 3) sin(A-B) = sin A cos B - cos A sin B 4) cos(A+B) = cos A cos B - sin A sin B 5) cos(A-B) = cos A cos B + sin A sin B

6) tan(A+B) = tan tan1 tan tan

A B

A B

7) tan(A-B) = tan tan1 tan tan

A B

A B

8) cot(A+B) = cot cot 1cot cot

A B

B A

9) cot(A-B) = cot cot 1cot cot

A B

B A

10) sin 2A = 2sin A sin A 11) cos 2A = cos2A sin2A 12) cos 2A = 2cos2A 1 13) cos 2A = 1 2sin A

14) sin A cos B = sin(A+B) + sin(A-B)2

15) cos A sin B = sin(A+B) - sin(A-B)2

16) cos A cos B = cos(A+B) + cos(A-B)2

17) sin A sin B = cos(A-B) - cos(A+B)2

20

18) sin A + sin B = 2sin2

A B

cos2

A B

19) sin A - sin B = 2sin2

A B

cos2

A B

20) cos A + cos B = 2cos2

A B

cos2

A B

21) cos A - cos B = 2sin2

A B

sin2

A B

3. 3.1 1) arcsin

arcsin B = A sin A = B 2) arccos

arccos B = A cos A = B 3) arctan

arctan B = A tan A = B 4) arcsec

arcsec B = A sec A = B 5) arccosec

arccosec B = A cosec A = B 6) arccot

arccot B = A cot A = B

21

3.2 1) sin(arcsin x) = x -1 x 1 2) arcsin(sin x) = x

2

x

2

3) cos(arccos x) = x -1 x 1 4) arccos(cos x) = x 0 x 5) tan(arctan x) = x x 6) arctan(tan x) = x

2

x

2

4.

ABC A, B , C a, b, c

a2 = b2 + c2 2bc cos A

b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C

sin Aa

= sin Bb

= sinCc

ABC = 12 ab sin C = 1

2 ac sin B = 1

2 bc sin A

A B

C

a b

c

22

1. ABC sin A = 35

cos B = 513

cos C

2. cot(arccot 7 + arccot 13 + arccot 21 + arc 31)

3. ABC A, B C a, b c a2 = b2 = 31c2 3tan C(cot A + cot B)

4. A cos x = cos (4

x ) A(0 , 24)

23

5. 6. 7.

24

8.

9.

10.

25

11. 12.

26

13. 14. 15.

27

16. 17. 18.

28

19. 20.

trigonometric.pdf

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