sin = sin cos + cos sin sin cos - cos sin 2 cos = cos cos - sin sin cos = cos cos...

14
ALG 3/TRIG(10) 1 1 ADDITION AND SUBTRACTION FORMULAS sin α + = sin cos + cos sin sin α - = sin cos - cos sin cos α + = cos cos - sin sin cos α - = cos cos + sin sin tan α + = tan tan 1 tan tan tan α - = tan tan 1 tan tan COMBINATIONS 15 = 345 = 255 = 75 = EXAMPLES Evaluate each expression 1) sin 75 2) cos 345 0, R 2 2 3 2 30 60 1 45 45 1 C A S T

Transcript of sin = sin cos + cos sin sin cos - cos sin 2 cos = cos cos - sin sin cos = cos cos...

ALG 3/TRIG(10) 1

1

ADDITION AND SUBTRACTION FORMULAS sin α + = sin cos + cos sin

sin α - = sin cos - cos sin

cos α + = cos cos - sin sin

cos α - = cos cos + sin sin

tan α + = tan tan

1 tan tan

tan α - = tan tan

1 tan tan

COMBINATIONS

15 = 345 =

255 = 75 =

EXAMPLES Evaluate each expression 1) sin 75

2) cos 345

0 ,R

2

2

3

2

30

60

1

45

45

1

C

A S

T

ALG 3/TRIG(10) 2

2

3) tan 165

Simplify the following: 4) cos (270 - x)

5) sin ( x + 2

) =

6) cos ( R

π + x )

ALG 3/TRIG(10) 3

3

Find each of the following numbers , please.

If sin A = 12

13 , 0 < A <

2 and cos B =

8

17 ,

3

2B

7) sin (A + B) 8) cos (A – B) 9) tan (A + B ) 10) 7) Verify the identity: Only work one side.

(a) π 2

cos θ = cosθ sinθ4 2

(b) cos θ 30 cos θ 30 = 3cosθ

ALG 3/TRIG(10) 4

4

Algebra 3 Trig Formulas Assignment #4

(1) Find each of the following numbers please.

(a) sin(15

) (b) cos(15

)

© sin(105

) (d) cos(75

)

(e) sin(195

) (f) cos(165

)

(g) sin(345

) (h) tan(15

)

(2) Simplify each of the following please.

(a) sin(90

+ x) (b) cos(90

x)

© sin(180

x) (d) cos(180

+ x)

(3) Sin(A) = 5

4 ,

20 < A < , Cos(B) =

13

5 ,

2 < B < . Find each of the following numbers

please.

(a) sin(A + B) (b) cos(A + B)

© sin(A B) (d) cos(A B)

(e) tan(A + B) (f) csc(A B)

(4) Use addition or subtraction formulas to verify the identity

(a) π 1 tanθ

tan θ = 4 1 tanθ

(b) π 1

sin θ = cosθ 3sinθ6 2

ALG 3/TRIG(10) 5

5

Assignment #4

Answers

(1) (a) 4

2 6 (b)

4

2 6

(c) 4

2 6 (d)

4

2 6

(e) 2 6

4 (f)

4

2 6

(g) 4

6 2 (h) 2 3

(2) (a) cos x (b) sin x

(c) sin x (d) cos x

(3) (a) 1665

(b) - 6563

(c) - 6556 (d) 33

65

(e) 6316 (f) 65

56

ALG 3/TRIG(10) 6

6

DOUBLE AND HALF ANGLE FORMULAS

Double – Angle Formulas Half – Angle Formulas

2

2

2

2

2

sin

sin

sin

sin 2 = 2

cos 2 = -

1 - 2

= 2 1

2tantan2 =

1

cos

t

o

cos

an

c s

1 sincos tan =

2 2 2 1 cos

1 1 cossin tan =

2 2 2

cos

c s

sin

o

Find each of the following numbers, please.

1) sin ( 1

222

)

2) cos (1

1572

)

If Sin A = 5

13, R3

A2

Tan B = R3, B

4 2

Find the following numbers, please.

3) sin (1

2A)

4) cos (2B)

ALG 3/TRIG(10) 7

7

5) sin (A + B)

6) If is obtuse such that the 5

sin = 13

, find the sin 2

7) A photographer wants to take a picture of a 4’ vase standing on a 3’ pedestal. She wants to position the camera at a point C on the floor so that the angle of elevation to the top of the pedestal and the angle from the bottom of the vase to the top of the vase are the same. How far away from the foot of the pedestal should the camera be placed?

8) Evaluate: 2 21 3π 3π

cos sin4 4 4

15

15

ALG 3/TRIG(10) 8

8

Algebra 3 Double and Half Angle Formulas Assignment #5

(1) Find each of the following numbers please.

(a) sin(6721

) (b) cos(2221

)

(c) sin(11221

) (d) cos(20221

)

(2) Sin(A) = 5

4 ,

2 < A < , Tan(B) =

5

12 , 3

2 < B < 2 . Find each of the following

numbers please.

(a) sin(21 A) (b) cos(

21 A)

(c) sin(21 B) (d) cos(

21 B)

(e) sin(2B) (f) cos(2A)

(g) sin(A B) (h) cos(A + B)

(3) Evaluate

(a) 2 7π

1 sin12

(b) 2

6tan75

1 tan 75

(4) 2

If tan θ = and cos θ > 0, find the sin2θ.3

ALG 3/TRIG(10) 9

9

Identities

P. 458 # 29 – 39

P. 470 # 23,25,26,27,33,36,38,40,41

ALG 3/TRIG(10) 10

10

Answers

(1) (a) 2

2 2 (b)

2

2 2

(c) 2

2 2 (d)

2

2 2

(2) (a) 5

2 (b) 5

1

(c) 13

2 (d) 13

3

(e) 169120 (f)

257

(g) 6516 (h)

6533

(3) (a) 2 3

4

(b) 2

(4) 12

13

ALG 3/TRIG(10) 11

11

Alg 3 Formulas Review Assign #6 (1) Find each of the following numbers please.

(a) sin(75

) (b) cos(105

)

(c) sin 12

13 (d) cos 12

11

(e) sin(195

) (f) cos(285

)

(g) sin

21112 (h) cos

21157

(i) sin 8

11 (j) cos 8

3

(k) 2 sin

217 cos

217 (l) cos

24

2 sin24

2

(m) tan(75 ) (n) cos(60 )

(2) Simplify each of the following please.

(a) sin(180

+ x) (b) cos(90

x) (c) sin(270

x)

(d) cos(180

x) (e) sin x 2

(f) cos x 2

3

(3) Sin(A) = 5

4 ,

23 A , Sec(B) =

5

13 , 0 B

2. Find each of the following

numbers please.

(a) sin(A + B) (b) cos(A + B) (c) sin(A B)

(d) cos(A B) (e) sin(2A) (f) cos(2A)

(g) sin(2B) (h) cos(2B) (i) sin 2A

(j) cos 2A (k) sin

2B (l) cos

2B

ALG 3/TRIG(10) 12

12

Review Answers

(1) (a) 4

2 6 or

2

3 2 (b)

4

6 2 or

2

3 2

(c) 4

6 2 or

2

3 2 (d)

4

2 6 or

2

3 2

(e) 4

6 2 or

2

3 2 (f)

4

2 6 or

2

3 2

(g) 2

2 2 (h)

2

2 2

(i) 2

2 2 (j)

2

2 2

(k) 4

2 6 or

2

3 2 (l)

4

2 6 or

2

3 2

(m) 3 2 (n) 2

1

(2) (a) sin x (b) sin x (c) cos x

(d) cos x (e) cos(x) (f) sin(x)

(3) (a) 6516 (b)

6563 (c)

6556

(d) 6533 (e)

2524 (f)

257

(g) 169120 (h)

169119 (i)

5

2

(j) 5

1 (k) 13

2 (l) 13

3

ALG 3/TRIG(10) 13

13

7-4 through 7-5 Extra Review

1. Find each angle using angle addition/subtraction formulas a) sin 195º b) cos 165º 2. Determine the following

a) 21 2sin12

b) 1

sin15 cos152

c) cos75 d) 5

sin8

3. 4 5

cos (sin >0) and sin (tan <0).5 6

Find the following. A/2 and B/2 are in Q I

a) sin( ) b) cos( ) c) cos2 d) sin2

e) cos2

f) sin2

4. Find h:

h

50

Θ 2Θ

150

ALG 3/TRIG(10) 14

14

ANSWERS

6 2 2 61. b)

4 4

2 31 6 22. a) cos30 b) c) or +

8 4 2

2+ 2d)

2

3 11 20 4 11 15 73. a) b) c)

30 30 25

24 18 3 11d) e)

25 6

18 3 11 f)

6

2

2

2 2

2

2

2 2

2 tan5. tan tan 2

150 50 1 tan

75 150 50 150 75

1150

21 1

150 3 150

h h

h

h h h h

h

h h hh

2

22

2

3

15050 3

3h