1 algebra 3 assignment sheet - pequannock township high...

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Algebra 3 Assignment Sheet

WELCOME TO TRIGONOMETRY, ENJOY YOUR STAY

(1) Assignment # 1 − Complete the circle diagram

(2) Assignment # 2 − Sine & Cosine functions chart

(3) Assignment # 3 − Other Trig functions chart

(4) Assignment # 4 − Finding other trig functions

(5) Assignment # 5 − Review Worksheet

(6) TEST

(7) Assignment # 6 − Angle Addition Formulas

(8) Assignment # 7 − Double, Half-Angle Formulas

(9) Assignment # 8 − Review Worksheet

(10) TEST

(11) Assignment # 9 − Trig Identities (1)

(11) Assignment # 9 − Trig Identities (2)

(12) Assignment # 10 Problems ( 1 – 9 ) − Solving Trig Equations

(13) Assignment # 10 Problems ( 10 – 18 ) − Solving Trig Equations

(14) Assignment # 11 − Review Worksheet – Solving Trig Equations

(15) TEST

2

INTRODUCTION TO TRIGONOMETRY

I Definition of radian: Radians are an angular measurement. One radian is the measure of a central angle of a circle that is subtended by an arc whose length is equal to the radius of the circle. Therefore: arc length = angle in radians x radius The radius wraps itself around the circle 2π times. Approx. 6.28 times. Therefore 360o = 2π R

Dividing you get 21360

Rπ=o o ………..1

180

Rπ=o o

Conversely ………………………... 1801R Rπ=

o

Ex. Change 60o to radians.

Change 4

Rπ to degrees.

Convert the following from degrees to radians or vice versa: 1. 36o 2. 320o 3. 195 0

4. 15π 5. 17

20π 6. 5

3π

0 ,R

2π

2π

π

32π

1 R

5

5 2 R

5

10

1

2

3

6

4

5

3

II UNIT CIRCLE: The unit circle is the circle with radius = 1, center is located at the origin. What is the equation of this circle? Important Terms: A. Ini t ia l s ide : B. Terminal s ide : C. Coterminal angles : D. Reference angles : The initial and terminal sides form an angle at the center if the terminal side rotates CCW, the angle is pos i t ive if the terminal side rotates CW, the angle is negat ive unit circle positive negative coterminal

Coterminal angles have the same terminal side…. -45˚ and 315˚ or 3π and 7

3π

The reference angle is the acute angle made between the Terminal Side and the x-axis

4

III GEOMETRY REVIEW 30 – 60 – 90 ° RIGHT TRIANGLES 45 – 45 - 90 ° Therefore, for the Unit Circle, hypotenuse is always 1.

60°

30°

a 2a

3a

60°

30°

1/2 1

3 / 2

45°

45°

a

a

a 2

45°

45°

2 / 2

2 / 2

1

5

Algebra 3 Assignment # 1

6

Trigonometric Functions Let � “theta” represent the measure of the reference angle. Three basic functions are sine, cosine and tangent. They are written as sin �, cos �, and tan � Right triangle trigonometry - SOHCAHTOA

sin opphyp

θ =

cos adjhyp

θ =

tan oppadj

θ =

A. Find cos � B. Find sin � C. Find tan � D. Find sin �

θ

hyp

adj

opp

12 θ

5 13

θ

5

5 2

5 3

θ

6 9

θ

7 25

7

Triangles in the Unit Circle On the Unit Circle: I Where functions are positive II Reference Triangles A. Drop ⊥ from point to x-axis.

1

O

P(x,y)

A (1,0)

B (0,1)

x

y

=

=

=

sinθ

cosθ

tanθ

O

8

B. Examples

1. Find sin 34π⎛ ⎞

⎜ ⎟⎝ ⎠

2. Find cos 34π⎛ ⎞−⎜ ⎟

⎝ ⎠ Same as cos 5

4R⎛ ⎞

⎜ ⎟⎝ ⎠

π

3. Find sin 420° =

4. Find cos 136π⎛ ⎞−⎜ ⎟

⎝ ⎠ =

5. Find sin π = cos π =

coterminal angles

coterminal angles

coterminal angles

9

III Quadrangle Angles Def: An angle that has its terminal side on one of the coordinate axes. To find these angles , use the chart Find the sine, cosine for all the quadrangles. 0 0 sin 0 cos0

90 sin cos2

180 sin cos

3270 sin cos2

360 2 sin cos

R

R

R

R

R

π

π

π

π

° = = =

° = = =

° = = =

° = = =

° = = =

Trig values

A (1,0)

B (0,1)

C (-1,0)

D (0,-1)

1

1sincos

y

x

= =

= =

= =

sinθ y

cosθ x

ytanθx

10


Complete each of the following tables please.

Radian Measure

113π

54π

196π

3π

Degree Measure 330o 450o 45− o 210− o

Sin

Cos

Radian Measure

32π

− 73π

− 4π

114π

Degree Measure 180− o 150o 780o 90o

Sin

Cos

11

Answers

Radian Measure

113π

116π

54π

52π

196π

4π

− 3π 76π

−

Degree Measure 660o 330o 225o 450o 570o 45− o 540o 210− o

Sin 32

− 12

− 22

− 1 12

− 22

− 0 12

Cos 12

32

22

− 0 32

− 22

−1 32

−

Radian Measure

32π

− −π 73π

− 56π

4π

133π

114π

2π

Degree Measure 270− o 180− o 420− o 150o 45o 780o 495o 90o

Sin 1 0 32

− 12

22

32

22

1

Cos 0 −1 12

32

− 22

12

22

− 0

12

6.2 Other Trigonometric Functions

Sinθ Cosecant: Cosθ Secant: Tanθ Cotangent: http://mathplotter.lawrenceville.org/mathplotter/mathPage/trig.htm Find the following values

1. cscπ4

2. cotπ6

3. sec2π 4. sec3π2

5. tan 4π3

6. cot − π4

⎛ ⎝

⎞ ⎠ 7. csc3π 8. tan 17π

6

13

6.2 Algebra 3 Assignment # 3

Complete the following tables.

Radian Measure 3

8 π 4

3 π 6

π π5

Degree Measure 330° 450° −135° 240°

Sin

Cos

Tan

Cot

Sec

Csc

Radian Measure 2

3π−

37π

− 413π

47π

Degree Measure 540° 150° −210° 270°

Sin

Cos

Tan

Cot

Sec

Csc

Alg 3(11) 14 Ch 6 Trig

Answers

Radian Measure 3

8 π 6

11 π 4

3 π 25π

6 π

43 π

− π5 3

4 π

Degree Measure 480° 330° 135° 450° 30° −135° 900° 240°

Sin 23 −

21

22 1

21 −

22 0 −

23

Cos −21

23 −

22 0

23 −

22 −1 −

21

Tan − 3 − 13

−1 Undef. 13

1 0 3

Cot − 13

− 3 −1 0 3 1 Undef. 13

Sec −2 23

− 2 Undef. 23

− 2 −1 −2

Csc 23

−2 2 1 2 − 2 Undef. − 23

Radian Measure 2

3π− π3

37π

− 6

5 π 413π

67 π

− 47π

23π

Degree Measure −270° 540° −420° 150° 585° −210° 315° 270°

Sin 1 0 −23

21 −

22

21 −

22 −1

Cos 0 −1 21 −

23 −

22 −

23

22 0

Tan Undef. 0 − 3 − 13

1 − 13

−1 Undef.

Cot 0 Undef. − 13

− 3 1 − 3 −1 0

Sec Undef. −1 2 − 23

− 2 − 23

2 Undef.

Csc 1 Undef. − 23

2 − 2 2 − 2 −1

Alg 3(11) 15 Ch 6 Trig 6.2 MORE TRIG FUNCTIONS Identifying in which quadrant the angle lies is essential for having the correct signs of the trig functions.

If given, Sin � = 35

and if told that 0 ≤ θ ≤ π2

, can we find the cos �?

1. Find cosθ if sinθ = 2/3 and 0 ≤ θ ≤ π2

2. Find tanθ if sinθ = 3/7 and π2≤ θ ≤ π

θ 1 θ

θ

x


3. Find cscθ if cosθ = 32

− and π ≤ θ ≤3π2

4. Find secθ if sinθ = -1/3 and 3π2≤ θ ≤ 2π

5. If Tan θ = 4-5

, 270 θ<360° < ° , find all the remaining functions of θ.

6. Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (-5, -12) on its terminal ray.



(1) Sin(θ ) = 53

, 20 < < πθ . Find the remaining 5 trig. functions of θ .

(2) Cos(θ ) = 54

− , 2 < < π θ π. Find the remaining 5 trig. functions of θ .

(3) Tan(θ ) = 512

, 32 < < ππ θ . Find the remaining 5 trig. functions of θ .

(4) Sec(θ ) = 57

, 32 < < 2π θ π . Find the remaining 5 trig. functions of θ .

(5) Csc(θ ) = 37

− , 180 < < 270θo o. Find the remaining 5 trig. functions of θ .

(6) Cot(θ ) = 2− , 270 < < 360θo o. Find the remaining 5 trig. functions of θ .

(7) Sin(θ ) = 2524

− , 180 < < 270θo o. Find the remaining 5 trig. functions of θ .

(8) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (4 , −3) on

its terminal ray (9) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point (−5 , 12) on

its terminal ray


Trig Assignment #4 Answers

(1) cos(θ ) = 54 , tan(θ ) = 4

3 , cot(θ ) = 34 , sec(θ ) = 4

5 , csc(θ ) = 35

(2) sin(θ ) = 53 , tan(θ ) = − 4

3 , cot(θ ) = − 34 , sec(θ ) = − 4

5 , csc(θ ) = 35

(3) sin(θ ) = 1312− , cos(θ ) =

135− , cot(θ ) =

125 , sec(θ ) =

513− , csc(θ ) =

1213−

(4) sin(θ ) = 762− , cos(θ ) = 7

5 , tan(θ ) = 562− , cot(θ ) = 5

2 6− , csc(θ ) = 7

2 6−

(5) sin(θ ) = 73− , cos(θ ) =

7102− , tan(θ ) = 3

2 10 , cot(θ ) =

3102 , sec(θ ) = 7

2 10−

(6) sin(θ ) = 15

− , cos(θ ) = 25

, tan(θ ) = 21− , sec(θ ) =

25 , csc(θ ) = 5−

(7) cos(θ ) = 257− , tan(θ ) =

724 , cot(θ ) =

247 , sec(θ ) =

725− , csc(θ ) =

2425−

(8) sin(θ ) = 3

5− cos(θ ) = 54 , tan(θ ) = 3

4− , cot(θ ) = 43− , sec(θ ) = 4

5 , csc(θ ) = 53−

(9)sin(θ ) = 1213 cos(θ ) = 5

13− , tan(θ ) = 125− , cot(θ ) = 5

12− , sec(θ ) = 135− , csc(θ ) = 1312


Algebra 3 Review Worksheet (1) Complete the following table please.

Rad. 32π 2

3π π− 3 310π 4

9π− 4π−

Deg. 135° 330° −150° −750° 240°

sin

cos

tan

cot

sec

csc

(2) Sin(x) = 75 , π<<π x 2 . Find the remaining 5 trig functions of x.

(3) Tan(θ) = 21 , !! 90 0 <θ< . Find the remaining 5 trig functions of θ.

(4) Cot(x) = 0.8 , 2

3 x π<<π . Find the remaining 5 trig functions of x.

(5) Sec(θ) = −3 , !! 180 90 <θ< . Find the remaining 5 trig functions of θ. (6) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the

point (−5 , 3) on its terminal ray.


Algebra 3 Review Answers

(1)

Rad. 32π 4

3π 23π 6

11π π− 3 65π− 3

10π 625π− 4

9π− 34π 4

π−

Deg. 120° 135° 270° 330° −540° −150° 600° −750° −405° 240° −45°

sin 23

22 −1 2

1− 0 21− −

23 2

1− −22 −

23 −

22

cos 21− −

22 0

23 −1 −

23 2

1− 23

22 2

1− 22

tan 3− −1 Undef −13

0 13

3 - 13

−1 3 −1

cot −13

−1 0 3− Undef 3 13

3− −1 13

−1

sec −2 − 2 Undef 23

−1 −23

−2 23

2 −2 2

csc 23

2 −1 −2 Undef −2 −23

−2 − 2 −23

− 2

(2) cos(x) = 762− , tan(x) =

625− , cot(x) =

562− , sec(x) =

627− , csc(x) = 5

7

(3) sin(θ) = 51 , cos(θ) =

52 , cot(θ) = 2 , sec(θ) =

25 , csc(θ) = 5

(4) sin(x) =415− , cos(x) =

414− , tan(x) = 4

5 , sec(x) = 441− , csc(x) =

541−

(5) sin(θ) = 322 , cos(θ) = 3

1− , tan(θ) = 22− , cot(θ) = 221− , csc(θ) =

223

(6) sin(θ) = 343 , cos(θ) =

345− , tan(θ) = 5

3− , cot(θ) = 35− , sec(θ) =

534− , csc(θ) =

334

Alg 3(11) 21 Ch 6 Trig ADDITION AND SUBTRACTION FORMULAS sin ( )βα + = sinα cos β + cos α sinβ sin ( )βα - = sin α cos β - cosα sin β cos ( )βα + = cosα cosβ - sinα sin β cos ( )βα - = cosα cos β + sinα sin β

tan ( )βα + = tan tan1 tan tan

α βα β+

−

tan ( )βα - = tan tan1 tan tan

−+α βα β

SPECIAL ANGLES

30 45 60 90

6 4 3 2

° ° o o

π π π π

2nd 3rd 4th 120 ___ ___ 135 ___ ___ 150 ___ ___ 180 ___ ___ COMBINATIONS 15° = 345° =

255° = 512π

=

0 ,R

2π

2π

π

32π

Alg 3(11) 22 Ch 6 Trig EXAMPLES Evaluate each expression 1) sin 75 °

sin (45 + 30) sin(120 – 45) 2) cos 345°

3) tan 1112π

Alg 3(11) 23 Ch 6 Trig Simplify the following: 4) cos (270° - x)

5) sin ( x + 2π

) =

6) cos ( x + π )

Alg 3(11) 24 Ch 6 Trig Find each of the following numbers:

If sin A = 1213

, 0 < A < 2π

and cos B = 817

− , 32

Bπ π< <

7) sin (A + B) 8) cos (A – B) 9) tan (A + B )


Algebra 3 Trig Formulas Assignment #6

(1) Find each of the following numbers please.

(a) sin(15! ) (b) cos(15! )

(c) sin(105! ) (d) cos(75! )

(e) sin1312π⎛ ⎞

⎜ ⎟⎝ ⎠

(f) cos1112π⎛ ⎞

⎜ ⎟⎝ ⎠

(g) sin(345! ) (h) tan(15! )

(2) Simplify each of the following please.

(a) sin(90!+ x) (b) cos(2π− x)

(c) sin(180!− x) (d) cos(π+ x)

(3) Sin(A) = 54

, A is in Quadrant I, Cos(B) = −135

, B is in Quadrant II. Find each of the

following numbers please.

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

(c) sin(A − B) (d) cos(A − B)

(e) tan(A + B) (f) csc(A − B)


Assignment #6

Answers

(1) (a) 4

2 6 − (b)

42 6 +

(c) 4

2 6 + (d)

42 6 −

(e) 4

6 2 − (f)

42 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) 6365

−

(c) 5665

− (d) 3365

(e) 1663

− (f) 6556

−


DOUBLE AND HALF ANGLE FORMULAS

Double – Angle Formulas Half – Angle Formulas

2

2

2

2

sinsi

sn

in 2 = 2cos 2 = -

1 sin - 2 = 2 1

coscos

cos

θ

θ

=

θ

θ

θ

θ

θ−

θ

1cos2 2

1sin2

cos

cos2

θ += ±

θ −= ±

θ

θ

Find each of the following numbers, please.

1) sin ( °1222

)

2) cos (π78

)

C

A S

T


If Sin A = −513

, π

π < <3A2

Tan B = π

− < < π3, B4 2

Find the following numbers, please.

3) sin (12

A)

4) cos (2B)

5) sin (A + B)


Algebra 3 Double and Half Angle Formulas Assignment #7


(a) sin(67 21 ! ) (b) cos

8π⎛ ⎞

⎜ ⎟⎝ ⎠

(c) sin58π⎛ ⎞

⎜ ⎟⎝ ⎠

(d) cos(202 21 ! )

(2) Sin(A) = 54

, 2 < A < π π, Tan(B) = 512

− , 32 < B < 2π π . Find each of the following numbers

please.

(a) sin( 21A) (b) cos( 2

1A)

(c) sin( 21 B) (d) sec( 2

1 B)

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

(g) csc(A − B) (h) cos(A + B)


Answers

(1) (a) 2

2 2 + (b)

22 2 +

(c) 2

2 2 + (d) −

22 2 +

(2) (a) 25

(b) 15

(c) 213

(d) 133

−

(e) 169120− (f)

257−

(g) 6516− (h)

6533


Algebra 3 Formula Review Worksheet, Assignment #8


(a) sin(15! ) (b) cos(105! )

(c) sin(195! ) (d) cos(285! )

(e) sin(112 21 ! ) (f) cos

78π⎛ ⎞

⎜ ⎟⎝ ⎠

(g) tan(75°) (h) sec512π⎛ ⎞

⎜ ⎟⎝ ⎠

(2) Simplify each of the following please.

(a) sin(180!+ x) (b) cos(2π

+ x)

(c) sin(32π− x) (d) cos(180!− x)

(3) Sin(A) = −54

, 32 < A < ππ , Sec(B) =

513

, 20 < B < π . Find each of the following numbers.

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

(c) sin(A − B) (d) cos(A − B)

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

(g) sin ⎟⎠

⎞⎜⎝

⎛2A

(h) cos ⎟⎠

⎞⎜⎝

⎛2A


Answers

(1) (a) 4

2 6 − or

23 2 −

(b) 4

6 2 − or −

23 2 −

(c) 4

6 2 − or −

23 2 −

(d) 4

2 6 − or

23 2 −

(e) 2

2 2 + (f)

22 2 +

−

(g) 3 2 + (h) 6 2+

(2) (a) −sin ( )x (b) sin ( )x

(c) −cos ( )x (d) −cos ( )x

(3) (a) 6556− (b)

6533

(c) 6516 (d)

6563−

(e) 169120 (f)

257−

(g) 25

(h) 15

−

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