TRIGONOMETRY - Pingry Schoolfaculty.pingry.org/bpoprik/documents/TrigPacket1_000.doc · Web...

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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

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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 times. Approx. 6.28 times.Therefore = Dividing you get ………..

Conversely ………………………...

Ex. Change to radians.

Change to degrees.

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

0 ,R2

2

32

1 R55 2 R5

10

1

2

3

6

4

5

3

4. 5. 6.

4

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 positiveif 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 and

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

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III GEOMETRY REVIEW

30 – 60 – 90 RIGHT TRIANGLES 45 – 45 - 90

Therefore, for the Unit Circle, hypotenuse is always 1.

60

30

a2a

3a

60

30

1/21

3 / 2

45

45

a

a

a

45

45

2 / 2

2 / 2

1

7

Algebra 3 Assignment # 1

8

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

A. Find cos θ B. Find sin θ

C. Find tan θ D. Find sin θ

θ

hyp

adj

opp

12θ

5

13

θ

5

5

5

θ

69

θ

725

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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

O

10

B. Examples1. Find sin

2. Find cos Same as cos

3. Find sin =

4. Find cos =

5. Find sin = cos =

coterminal angles

coterminal angles

coterminal angles

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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.

Trig values

A (1,0)

B (0,1)

C (-1,0)

D (0,-1)

13


Complete each of the following tables please.

Radian Measure

Degree Measure

Sin

Cos

Radian Measure

Degree Measure

Sin

Cos

14

Answers

Radian Measure

Degree Measure

Sin 1 0

Cos 0 1

Radian Measure

Degree Measure

Sin 1 0 1

Cos 0 1 0

15

6.2 Other Trigonometric Functions

Sin Cosecant:

Cos Secant:

Tan Cotangent:

http://mathplotter.lawrenceville.org/mathplotter/mathPage/trig.htm

Find the following values1. 2. 3. 4.

5. 6. 7. 8.

http://mathplotter.lawrenceville.org/mathplotter/mathPage/trig.htm

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6.2 Algebra 3 Assignment # 3

Complete the following tables.

Radian MeasureDegree

Measure 330 450 135 240

Sin

Cos

Tan

Cot

Sec

Csc


Measure 540 150 210 270

Sin

Cos

Tan

Cot

Sec

Csc

Alg 3(11) 17Ch 6 Trig

Answers


Measure 480 330 135 450 30 135 900 240

Sin 1 0

Cos 0 1

Tan 1 Undef. 1 0

Cot 1 0 1 Undef.

Sec 2 Undef. 1 2

Csc 2 1 2 Undef.


Measure 270 540 420 150 585 210 315 270

Sin 1 0 1

Cos 0 1 0

Tan Undef. 0 1 1 Undef.

Cot 0 Undef. 1 1 0

Sec Undef. 1 2 Undef.

Csc 1 Undef. 2 2 1


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 θ = and if told that , can we find the cos θ?

1. Find cos if sin = 2/3 and 2. Find tan if sin = 3/7 and

θ1θ

θ

x


3. Find csc if cos = and 4. Find sec if sin = -1/3 and

5. If Tan = , , find all the remaining functions of .

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



(1) Sin( ) = , . Find the remaining 5 trig. functions of .

(2) Cos( ) = , . Find the remaining 5 trig. functions of .

(3) Tan( ) = , . Find the remaining 5 trig. functions of .

(4) Sec( ) = , . Find the remaining 5 trig. functions of .

(5) Csc( ) = , . Find the remaining 5 trig. functions of .

(6) Cot( ) = , . Find the remaining 5 trig. functions of .

(7) Sin( ) = , . Find the remaining 5 trig. functions of .

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

Trig Assignment #4 Answers

Alg 3(11) 22Ch 6 Trig(1) cos( ) = , tan( ) = , cot( ) = , sec( ) = , csc( ) =

(2) sin( ) = , tan( ) = , cot( ) = , sec( ) = , csc( ) =

(3) sin( ) = , cos( ) = , cot( ) = , sec( ) = , csc( ) =

(4) sin( ) = , cos( ) = , tan( ) = , cot( ) = , csc( ) =

(5) sin( ) = , cos( ) = , tan( ) = , cot( ) = , sec( ) =

(6) sin( ) = , cos( ) = , tan( ) = , sec( ) = , csc( ) =

(7) cos( ) = , tan( ) = , cot( ) = , sec( ) = , csc( ) =

(8) sin( ) = cos( ) = , tan( ) = , cot( ) = , sec( ) = , csc( ) =

(9)sin( ) = cos( ) = , tan( ) = , cot( ) = , sec( ) = , csc( ) =

Algebra 3 Review Worksheet

(1) Complete the following table please.


Rad.

Deg. 135 330 150 750 240

sin

cos

tan

cot

sec

csc

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

(3) Tan() = , . Find the remaining 5 trig functions of .

(4) Cot(x) = 0.8 , . Find the remaining 5 trig functions of x.

(5) Sec() = 3 , . Find the remaining 5 trig functions of .

(6) Find the values of the six trig. functions of , if is an angle in standard position with the point (5 , 3) on its terminal ray.

Algebra 3 Review Answers(1)

Rad.


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

sin 1 0

cos 0 1

tan 1 Undef 0 - 1 1

cot 1 0 Undef 1 1

sec 2 Undef 1 2 2

csc 1 2 Undef 2 2

(2) cos(x) = , tan(x) = , cot(x) = , sec(x) = , csc(x) =

(3) sin() = , cos() = , cot() = 2 , sec() = , csc() =

(4) sin(x) = , cos(x) = , tan(x) = , sec(x) = , csc(x) =

(5) sin() = , cos() = , tan() = , cot() = , csc() =

(6) sin() = , cos() = , tan() = , cot() = , sec() = , csc() =


ADDITION AND SUBTRACTION FORMULAS

sin = sin cos + cos sinsin = sin cos - cos sincos = cos cos - sin sincos = cos cos + sin sin

tan =

tan =

SPECIAL ANGLES

2 nd 3 rd 4 th

120 ___ ___

135 ___ ___

150 ___ ___

180 ___ ___

COMBINATIONS

15 = 345 =

255 = =

0 ,R2

2

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Alg 3(11) 26Ch 6 TrigEXAMPLES

Evaluate each expression

1) sin 75

sin (45 + 30) sin(120 – 45)

2) cos 345

3) tan


Simplify the following:

4) cos (270 - x)

5) sin ( x + ) =

6) cos ( )

Alg 3(11) 28Ch 6 TrigFind each of the following numbers:

If sin A = , 0 < A < and cos B = ,

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) sin (f) cos

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

(2) Simplify each of the following please.

(a) sin(90 + x) (b) cos( x)

(c) sin(180 x) (d) cos( + x)

(3) Sin(A) = , A is in Quadrant I, Cos(B) = , 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) (b)

(c) (d)

(e) (f)

(g) (h)

(2) (a) cos (b) sin

(c) sin (d) cos

(3) (a) (b)

(c) (d)

(e) (f)


DOUBLE AND HALF ANGLE FORMULAS

Double – Angle Formulas Half – Angle Formulas

Find each of the following numbers, please.

1) sin ( )

2) cos ( )

C

AS

T


If Sin A = , Tan B =

Find the following numbers, please.

3) sin ( A)

4) cos (2B)

5) sin (A + B)

Alg 3(11) 33Ch 6 TrigAlgebra 3 Double and Half Angle Formulas Assignment #7


(a) sin(67 ) (b) cos

(c) sin (d) cos(202 )

(2) Sin(A) = , , Tan(B) = , . Find each of the following numbers please.

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

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

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

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


Answers

(1) (a) (b)

(c) (d)

(2) (a) (b)

(c) (d)

(e) (f)

(g) (h)


Algebra 3 Formula Review Worksheet, Assignment #8


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

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

(e) sin(112 ) (f) cos

(g) tan(75) (h) sec

(2) Simplify each of the following please.

(a) sin(180 + x) (b) cos( + x)

(c) sin( x) (d) cos(180 x)

(3) Sin(A) = , , Sec(B) = , . 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 (h) cos


Answers

(1) (a) or (b) or

(c) or (d) or

(e) (f)

(g) (h)

(2) (a) sin (b) sin

(c) cos (d) cos

(3) (a) (b)

(c) (d)

(e) (f)

(g) (h)

TRIGONOMETRY - Pingry Schoolfaculty.pingry.org/bpoprik/documents/TrigPacket1_000.doc · Web...

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