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
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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−
Related Documents
