TRIGONOMETRY - Pingry School



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

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 [pic] times. Approx. 6.28 times.

Therefore [pic] = [pic][pic]

Dividing you get [pic]………..[pic]

Conversely ………………………...[pic]

Ex. Change [pic] to radians.

Change [pic]to degrees.

Convert the following from degrees to radians or vice versa:

1. [pic] 2. [pic] 3. 195[pic]

4. [pic] 5. [pic] 6. [pic]

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 [pic] and [pic]

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

III GEOMETRY REVIEW

30 – 60 – 90[pic] RIGHT TRIANGLES 45 – 45 - 90[pic]

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

Algebra 3 Assignment # 1

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

[pic]

[pic]

[pic]

A. Find cos θ B. Find sin θ

C. Find tan θ D. Find sin θ

Triangles in the Unit Circle

On the Unit Circle:

I

Where functions are positive

II Reference Triangles

A. Drop [pic]from point to x-axis.

B. Examples

1. Find sin [pic]

2. Find cos[pic] Same as cos[pic]

3. Find sin [pic] =

4. Find cos [pic] =

5. Find sin [pic] = cos [pic] =

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.

[pic]

Trig values

Algebra 3 Assignment # 2

Complete each of the following tables please.

Radian Measure |[pic] | |[pic] | |[pic] | |[pic] | | |Degree Measure | |[pic] | |[pic] | |[pic] | |[pic] | |Sin | | | | | | | | | |Cos | | | | | | | | | |

Radian Measure |[pic] | |[pic] | |[pic] | |[pic] | | |Degree Measure | |[pic] | |[pic] | |[pic] | |[pic] | |Sin | | | | | | | | | |Cos | | | | | | | | | |

Answers

Radian Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Degree Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Sin |[pic] |[pic] |[pic] |1 |[pic] |[pic] |0 |[pic] | |Cos |[pic] |[pic] |[pic] |0 |[pic] |[pic] |(1 |[pic] | |

Radian Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Degree Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Sin |1 |0 |[pic] |[pic] |[pic] |[pic] |[pic] |1 | |Cos |0 |(1 |[pic] |[pic] |[pic] |[pic] |[pic] |0 | |

6.2 Other Trigonometric Functions

Sin( Cosecant:

Cos( Secant:

Tan( Cotangent:



Find the following values

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

6.2 Algebra 3 Assignment # 3

Complete the following tables.

Radian Measure |[pic] | |[pic] | |[pic] | |[pic] | | |Degree Measure | |330( | |450( | |(135( | |240( | |Sin | | | | | | | | | |Cos | | | | | | | | | |Tan | | | | | | | | | |Cot | | | | | | | | | |Sec | | | | | | | | | |Csc | | | | | | | | | |

Radian Measure |[pic] | |[pic] | |[pic] | |[pic] | | |Degree Measure | |540( | |150( | |(210( | |270( | |Sin | | | | | | | | | |Cos | | | | | | | | | |Tan | | | | | | | | | |Cot | | | | | | | | | |Sec | | | | | | | | | |Csc | | | | | | | | | |

Answers

Radian Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Degree Measure |480( |330( |135( |450( |30( |(135( |900( |240( | |Sin |[pic] |([pic] |[pic] |1 |[pic] |([pic] |0 |([pic] | |Cos |([pic] |[pic] |([pic] |0 |[pic] |([pic] |(1 |([pic] | |Tan |([pic] |([pic] |(1 |Undef. |[pic] |1 |0 |[pic] | |Cot |([pic] |([pic] |(1 |0 |[pic] |1 |Undef. |[pic] | |Sec |(2 |[pic] |([pic] |Undef. |[pic] |([pic] |(1 |(2 | |Csc |[pic] |(2 |[pic] |1 |2 |([pic] |Undef. |([pic] | |

Radian Measure |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Degree Measure |(270( |540( |(420( |150( |585( |(210( |315( |270( | |Sin |1 |0 |([pic] |[pic] |([pic] |[pic] |([pic] |(1 | |Cos |0 |(1 |[pic] |([pic] |([pic] |([pic] |[pic] |0 | |Tan |Undef. |0 |([pic] |([pic] |1 |([pic] |(1 |Undef. | |Cot |0 |Undef. |([pic] |([pic] |1 |([pic] |(1 |0 | |Sec |Undef. |(1 |2 |([pic] |([pic] |([pic] |[pic] |Undef. | |Csc |1 |Undef. |([pic] |2 |([pic] |2 |([pic] |(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 = [pic] and if told that [pic], can we find the cos θ?

1. Find cos( if sin( = 2/3 and [pic] 2. Find tan( if sin( = 3/7 and [pic]

3. Find csc( if cos( = [pic] and [pic] 4. Find sec( if sin( = -1/3 and [pic]

5. If Tan ( = [pic] , [pic], 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.

Algebra 3 Assignment # 4

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

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

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

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

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

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

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

(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

(1) cos([pic]) = [pic] , tan([pic]) = [pic] , cot([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

(2) sin([pic]) = [pic] , tan([pic]) = ([pic] , cot([pic]) = ([pic] , sec([pic]) = ([pic] , csc([pic]) = [pic]

(3) sin([pic]) = [pic] , cos([pic]) = [pic] , cot([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

(4) sin([pic]) = [pic] , cos([pic]) = [pic] , tan([pic]) = [pic] , cot([pic]) = [pic] , csc([pic]) = [pic]

(5) sin([pic]) = [pic] , cos([pic]) = [pic] , tan([pic]) = [pic] , cot([pic]) = [pic] , sec([pic]) = [pic]

(6) sin([pic]) = [pic] , cos([pic]) = [pic] , tan([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

(7) cos([pic]) = [pic] , tan([pic]) = [pic], cot([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

(8) sin([pic]) = [pic] cos([pic]) = [pic] , tan([pic]) = [pic] , cot([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

(9)sin([pic]) = [pic] cos([pic]) = [pic] , tan([pic]) = [pic] , cot([pic]) = [pic] , sec([pic]) = [pic] , csc([pic]) = [pic]

Algebra 3 Review Worksheet

(1) Complete the following table please.

Rad. |[pic] | |[pic] | |[pic] | |[pic] | |[pic] | |[pic] | |Deg. | |135( | |330( | |(150( | |(750( | |240( | | |sin | | | | | | | | | | | | |cos | | | | | | | | | | | | |tan | | | | | | | | | | | | |cot | | | | | | | | | | | | |sec | | | | | | | | | | | | |csc | | | | | | | | | | | | |

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

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

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

(5) Sec(() = (3 , [pic]. 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. |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |Deg. |120( |135( |270( |330( |(540( |(150( |600( |(750( |(405( |240( |(45( | |sin |[pic] |[pic] |(1 |[pic] |0 |[pic] |([pic] |[pic] |([pic] |([pic] |([pic] | |cos |[pic] |([pic] |0 |[pic] |(1 |([pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |tan |[pic] |(1 |Undef |([pic] |0 |[pic] |[pic] |-[pic] |(1 |[pic] |(1 | |cot |([pic] |(1 |0 |[pic] |Undef |[pic] |[pic] |[pic] |(1 |[pic] |(1 | |sec |(2 |([pic] |Undef |[pic] |(1 |([pic] |(2 |[pic] |[pic] |(2 |[pic] | |csc |[pic] |[pic] |(1 |(2 |Undef |(2 |([pic] |(2 |([pic] |([pic] |([pic] | |

(2) cos(x) = [pic] , tan(x) = [pic] , cot(x) = [pic] , sec(x) = [pic] , csc(x) = [pic]

(3) sin(() = [pic] , cos(() = [pic] , cot(() = 2 , sec(() = [pic] , csc(() = [pic]

(4) sin(x) =[pic] , cos(x) = [pic] , tan(x) = [pic] , sec(x) = [pic] , csc(x) = [pic]

(5) sin(() = [pic], cos(() = [pic] , tan(() = [pic] , cot(() = [pic] , csc(() = [pic]

(6) sin(() = [pic], cos(() = [pic], tan(() = [pic] , cot(() = [pic], sec(() = [pic] , csc(() = [pic]

ADDITION AND SUBTRACTION FORMULAS

sin [pic] = sin[pic]cos [pic] + cos [pic]sin[pic]

sin [pic] = sin [pic]cos[pic] - cos[pic]sin[pic]

cos [pic] = cos[pic]cos[pic] - sin[pic] sin[pic]

cos [pic] = cos[pic]cos[pic] + sin[pic]sin[pic]

tan [pic] = [pic]

tan [pic] = [pic]

SPECIAL ANGLES [pic]

2nd 3rd 4th

120 ___ ___

135 ___ ___

150 ___ ___

180 ___ ___

COMBINATIONS

15[pic] = 345[pic] =

255[pic] = [pic] =

EXAMPLES

Evaluate each expression

1) sin 75[pic]

sin (45 + 30) sin(120 – 45)

2) cos 345[pic]

3) tan [pic]

Simplify the following:

4) cos (270[pic] - x)

5) sin ( x + [pic]) =

6) cos ( [pic] )

Find each of the following numbers:

If sin A = [pic] , 0 < A < [pic] and cos B = [pic] , [pic]

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[pic]) (b) cos(15[pic])

(c) sin(105[pic]) (d) cos(75[pic])

(e) sin[pic] (f) cos[pic]

(g) sin(345[pic]) (h) tan(15[pic])

(2) Simplify each of the following please.

(a) sin(90[pic]+ x) (b) cos([pic]( x)

(c) sin(180[pic]( x) (d) cos([pic]+ x)

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

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

(g) [pic] (h) [pic]

(2) (a) cos[pic] (b) sin[pic]

(c) sin[pic] (d) (cos[pic]

(3) (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

DOUBLE AND HALF ANGLE FORMULAS

Double – Angle Formulas Half – Angle Formulas

[pic] [pic]

Find each of the following numbers, please.

1) sin ( [pic])

2) cos ([pic])

If Sin A = [pic], [pic][pic] Tan B = [pic]

Find the following numbers, please.

3) sin ([pic]A)

4) cos (2B)

5) sin (A + B)

Algebra 3 Double and Half Angle Formulas Assignment #7

(1) Find each of the following numbers please.

(a) sin(67[pic][pic]) (b) cos[pic]

(c) sin[pic] (d) cos(202[pic][pic])

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

(a) sin([pic]A) (b) cos([pic]A)

(c) sin([pic]B) (d) sec([pic]B)

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

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

Answers

(1) (a) [pic] (b) [pic]

(c) [pic] (d) ([pic]

(2) (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

(g) [pic] (h) [pic]

Algebra 3 Formula Review Worksheet, Assignment #8

(1) Find each of the following numbers please.

(a) sin(15[pic]) (b) cos(105[pic])

(c) sin(195[pic]) (d) cos(285[pic])

(e) sin(112[pic][pic]) (f) cos[pic]

(g) tan(75() (h) sec[pic]

(2) Simplify each of the following please.

(a) sin(180[pic]+ x) (b) cos([pic]+ x)

(c) sin([pic]( x) (d) cos(180[pic]( x)

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

Answers

(1) (a) [pic] or [pic] (b) [pic] or ([pic]

(c) [pic] or ([pic] (d) [pic] or [pic]

(e) [pic] (f) [pic]

(g) [pic] (h) [pic]

(2) (a) (sin[pic] (b) sin[pic]

(c) (cos[pic] (d) (cos[pic]

(3) (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

(g) [pic] (h) [pic]

-----------------------

[pic]

5

10

[pic]

5

5

O

[pic]

[pic]

[pic]

[pic]

[pic]

13

5

¸

[pic]

¸

6

[pic]

[pic]

[pic]

θ

6

[pic]

[pic]

[pic]

[pic]

A (1,0)

B (0,1)

C (-1,0)

D (0,-1)

5[pic]

y

x

B (0,1)

A (1,0)

P(x,y)

O

1

T

S

A

C

[pic]

[pic]

coterminal angles

coterminal angles

coterminal angles

[pic]

1

1/2

30[pic]

60[pic]

[pic]

[pic]

[pic]

45[pic]

45[pic]

[pic]

2a

a

30[pic]

60[pic]

a[pic]

a

a

45[pic]

45[pic]

opp

hyp

12

θ

adj

θ

5

5[pic]

9

θ

7

25

1

θ

θ

θ

x

5

4

6

3

2

1

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