1

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 # 9a− Trig Identities (1)

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

(12) Assignment # 10 Problems − Solving Trig Equations

(13) Assignment # 11 Problems − Solving Trig Equations

(14) Assignment # 12 − Review Worksheet

(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 360� = 2π R

Dividing you get 2

1360

Rπ=�

�

………..1180

Rπ=�

�

Conversely ………………………...180

1RRπ

=�

Ex. Change 60� to radians.

Change 4

Rπto degrees.

Convert the following from degrees to radians or vice versa:

1. 36� 2. 320� 3. 195 0

4. 15

π 5.

17

20

π 6.

5

3

π

0 ,R

2π

2

π

π

3

2

π

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. Initial side: B. Terminal side: 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 positive if the terminal side rotates CW, the angle is negative unit circle positive negative coterminal

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

π and

7

3

π

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 θ “theta” represent the measure of the reference angle. Three basic functions are sine, cosine and tangent. They are written as sin θ, cos θ, and tan θ

Right triangle trigonometry - SOHCAHTOA

sinopp

hypθ =

cosadj

hypθ =

tanopp

adjθ =

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

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 3

4π

2. Find cos3

4π −

Same as cos

5

4

R

π

3. Find sin 420° =

4. Find cos 13

6π −

=

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 cos 0

90 sin cos2

180 sin cos

3270 sin cos

2

360 2 sin cos

R

R

R

R

R

π

π

π

π

° = = =

° = = =

° = = =

° = = =

° = = =

A (1,0)

B (0,1)

C (-1,0)

D (0,-1)

1

1

sin

cos

y

x

= =

= =

= =

sinθ y

cosθ x

ytanθ

x

10

Algebra 3 Assignment # 2 Complete each of the following tables please.

Radian

Measure

11

3

π

5

4

π

19

6

π 3π

Degree

Measure 330� 450� 45− �

210− �

Sin

Cos

Radian

Measure

3

2

π−

7

3

π−

4

π

11

4

π

Degree

Measure 180− �

150� 780� 90�

Sin

Cos

11

Answers

Radian

Measure

11

3

π

11

6

π

5

4

π

5

2

π

19

6

π

4

π− 3π

7

6

π−

Degree

Measure 660� 330� 225� 450� 570� 45− �

540� 210− �

Sin 3

2−

1

2−

2

2− 1

1

2−

2

2− 0

1

2

Cos 1

2

3

2

2

2− 0

3

2−

2

2 −1

3

2−

Radian

Measure

3

2

π− −π

7

3

π−

5

6

π

4

π

13

3

π

11

4

π

2

π

Degree

Measure 270− �

180− �

420− �

150� 45� 780� 495� 90�

Sin 1 0 3

2−

1

2

2

2

3

2

2

2 1

Cos 0 −1 1

2

3

2−

2

2

1

2

2

2− 0

12

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. tan4π3

6. cot −π4

7. csc 3π 8. tan

17π6

13

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−

3

7π−

4

13π

4

7π

Degree

Measure 540° 150° −210° 270°

Sin

Cos

Tan

Cot

Sec

Csc

Alg 3(10) 14

Answers

Radian

Measure 3

8

π

6

11

π

4

3

π

2

5π

6

π

4

3

π− π5

3

4

π

Degree

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

Sin 2

3 −

2

1

2

2 1

2

1 −

2

2 0 −

2

3

Cos −2

1

2

3 −

2

2 0

2

3 −

2

2 −1 −

2

1

Tan − 3 − 1

3 −1 Undef.

1

3 1 0 3

Cot − 1

3 − 3 −1 0 3 1 Undef.

1

3

Sec −2 2

3 − 2 Undef.

2

3 − 2 −1 −2

Csc 2

3 −2 2 1 2 − 2 Undef. − 2

3

Radian

Measure 2

3π− π3

3

7π−

6

5

π

4

13π

6

7

π−

4

7π

2

3π

Degree

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

Sin 1 0 −2

3

2

1 −

2

2

2

1 −

2

2 −1

Cos 0 −1 2

1 −

2

3 −

2

2 −

2

3

2

2 0

Tan Undef. 0 − 3 − 1

3 1 − 1

3 −1 Undef.

Cot 0 Undef. − 1

3 − 3 1 − 3 −1 0

Sec Undef. −1 2 − 2

3 − 2 − 2

3 2 Undef.

Csc 1 Undef. − 2

3 2 − 2 2 − 2 −1

Alg 3(10) 15

MORE TRIG FUNCTIONS Identifying in which quadrant the angle lies is essential for having the correct signs of the trig functions.

If given, Sin θ = 3

5 and if told that 0 ≤ θ ≤

π2

, can we find the cos θ?

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

Alg 3(10) 16

3. Find cscθ if cosθ = 3

2

− 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.

Alg 3(10) 17

Algebra 3 Assignment # 4

(1) Sin(θ ) = 5

3 ,

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

(2) Cos( θ ) = 5

4− ,

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

(3) Tan( θ ) = 5

12 , 3

2 < < ππ θ . Find the remaining 5 trig. functions of θ .

(4) Sec( θ ) = 5

7 , 3

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

(5) Csc( θ ) = 3

7− , 180 < < 270θ� �

. Find the remaining 5 trig. functions of θ .

(6) Cot(θ ) = 2− , 270 < < 360θ� �

. Find the remaining 5 trig. functions of θ .

(7) Sin(θ ) = 25

24− , 180 < < 270θ� �

. 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

Alg 3(10) 18

Trig Assignment #4 Answers

(1) cos( θ ) = 54

, tan(θ ) = 43

, cot(θ ) = 34

, sec( θ ) = 45

, csc( θ ) = 35

(2) sin( θ ) = 53

, tan( θ ) = −43

, cot(θ ) = −34

, sec( θ ) = −45

, csc( θ ) = 35

(3) sin( θ ) = 1312− , cos( θ ) =

135− , cot(θ ) =

125

, sec( θ ) = 5

13− , csc(θ ) = 1213−

(4) sin( θ ) = 7

62− , cos( θ ) = 75

, tan(θ ) = 5

62− , cot( θ ) = 5

2 6− , csc( θ ) =

7

2 6−

(5) sin( θ ) = 73− , cos( θ ) =

7

102− , tan(θ ) = 3

2 10 , cot(θ ) =

3

102 , sec( θ ) =

7

2 10−

(6) sin( θ ) = 1

5− , cos( θ ) =

2

5 , tan( θ ) =

21− , sec( θ ) =

2

5 , csc( θ ) = 5−

(7) cos( θ ) = 257− , tan(θ ) =

724

, cot(θ ) = 247

, sec( θ ) = 725− , csc( θ ) =

2425−

(8) sin( θ ) = 35

− cos( θ ) = 54

, tan(θ ) = 34

− , cot( θ ) = 43

− , sec( θ ) = 45

, csc( θ ) = 53

−

(9)sin( θ ) = 1213

cos( θ ) = 513

− , tan(θ ) = 125

− , cot(θ ) = 512

− , sec( θ ) = 135

− , csc( θ ) = 1312

Alg 3(10) 19

Algebra 3 Review Worksheet

(1) Complete the following table please.

Rad. 3

2π 2

3π π− 3 3

10π 4

9π− 4π−

Deg. 135° 330° −150° −750° 240°

sin

cos

tan

cot

sec

csc

(2) Sin(x) = 7

5 , 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.

Alg 3(10) 20

Algebra 3 Review Answers (1)

Rad. 3

2π 4

3π 2

3π 6

11π π− 3 6

5π− 3

10π 6

25π− 4

9π− 3

4π 4π−

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

sin 23

22

−1 21− 0

21− −

23

21− −

22

−23

−22

cos 21− −

22

0 23

−1 −23

21−

23

22

21−

22

tan 3− −1 Undef −1

3 0

1

3 3 -

1

3 −1 3 −1

cot −1

3 −1 0 3− Undef 3

1

3 3− −1

1

3 −1

sec −2 − 2 Undef 2

3 −1 −

2

3 −2

2

3 2 −2 2

csc 2

3 2 −1 −2 Undef −2 −

2

3 −2 − 2 −

2

3 − 2

(2) cos(x) = 7

62− , tan(x) = 62

5− , cot(x) = 5

62− , sec(x) = 62

7− , csc(x) = 57

(3) sin(θ) = 5

1 , cos(θ) = 5

2 , cot(θ) = 2 , sec(θ) = 25 , csc(θ) = 5

(4) sin(x) =41

5− , cos(x) = 41

4− , tan(x) = 45 , sec(x) =

441− , csc(x) =

541−

(5) sin(θ) = 3

22 , cos(θ) = 31− , tan(θ) = 22− , cot(θ) =

22

1− , csc(θ) = 22

3

(6) sin(θ) = 34

3 , cos(θ) = 34

5− , tan(θ) = 53− , cot(θ) =

35− , sec(θ) =

534− , csc(θ) =

334

Alg 3(10) 21

ADDITION AND SUBTRACTION FORMULAS

sin ( )βα + = sinα cos β + cos α sinβ

sin ( )βα - = sin α cosβ - cosα sinβ

cos ( )βα + = cosα cosβ - sinα sinβ

cos ( )βα - = cosα cosβ + sinα sinβ

tan ( )βα + = tan tan

1 tan tan

α βα β+

−

tan ( )βα - = tan tan

1 tan tan

−+

α βα β

SPECIAL ANGLES

30 45 60 90

6 4 3 2

° ° � �

π π π π

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

255° = 5

12

π =

0 ,R

2π

2

π

π

3

2

π

Alg 3(10) 22

EXAMPLES Evaluate each expression 1) sin 75°

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

3) tan 11

12

π

Alg 3(10) 23

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

5) sin ( x + 2

π) =

6) cos ( x + π )

Alg 3(10) 24

Find each of the following numbers:

If sin A = 12

13 , 0 < A <

2

π and cos B =

8

17− ,

3

2Bπ π< <

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

Alg 3(10) 25

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) sin13

12

π

(f) cos11

12

π

(g) sin(345� ) (h) tan(15� )

(2) Simplify each of the following please.

(a) sin(90�+ x) (b) cos(2

π− x)

(c) sin(180�− x) (d) cos(π+ x)

(3) Sin(A) = 5

4 , A is in Quadrant I, Cos(B) = −

13

5 , 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)

Alg 3(10) 26

Assignment #6

Answers

(1) (a) 4

2 6 − (b)

4

2 6 +

(c) 4

2 6 + (d)

4

2 6 −

(e) 4

6 2 − (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) 16

65 (b)

63

65−

(c) 56

65− (d)

33

65

(e) 16

63− (f)

65

56−

Alg 3(10) 27

DOUBLE AND HALF ANGLE FORMULAS

Double – Angle Formulas Half – Angle Formulas

2

2

2

2

sin

si

s

n

in 2 = 2

cos 2 = -

1 sin - 2

= 2 1

cos

cos

cos

θ

θ

=

θ

θ

θ

θ

θ−

θ

1cos

2 2

1sin

2

cos

cos

2

θ += ±

θ −= ±

θ

θ

Find each of the following numbers, please.

1) sin ( °1

222

)

2) cos (π7

8)

C

A S

T

Alg 3(10) 28

If Sin A = −5

13,

ππ < <

3A

2 Tan B =

π− < < π

3, B

4 2

Find the following numbers, please.

3) sin (1

2A)

4) cos (2B)

5) sin (A + B)

Alg 3(10) 29

Algebra 3 Double and Half Angle Formulas Assignment #7

(1) Find each of the following numbers please.

(a) sin(6721 � ) (b) cos

8

π

(c) sin5

8

π

(d) cos(20221 � )

(2) Sin(A) = 5

4 ,

2 < A < π π , Tan(B) =

5

12− , 3

2 < B < 2π π . Find each of the following numbers

please.

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

21 A)

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

21 B)

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

(g) csc(A − B) (h) cos(A + B)

Alg 3(10) 30

Answers

(1) (a) 2

2 2 + (b)

2

2 2 +

(c) 2

2 2 + (d) −

2

2 2 +

(2) (a) 2

5 (b)

1

5

(c) 2

13 (d)

13

3−

(e) 169120− (f)

257−

(g) 6516

− (h) 6533

Alg 3(10) 31

Algebra 3 Formula Review Worksheet, Assignment #8

(1) Find each of the following numbers please.

(a) sin(15� ) (b) cos(105� )

(c) sin(195� ) (d) cos(285� )

(e) sin(1122

1 � ) (f) cos7

8

π

(g) tan(75°) (h) sec5

12

π

(2) Simplify each of the following please.

(a) sin(180�+ x) (b) cos(2

π+ x)

(c) sin(3

2

π− x) (d) cos(180�− x)

(3) Sin(A) = −5

4 , 3

2 < A < ππ , Sec(B) =

5

13 ,

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

2

A (h) cos

2

A

Alg 3(10) 32

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) 2

2 2 + (f)

2

2 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) 2

5 (h)

1

5−