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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
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
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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°
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
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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
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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
=
=
=
sinθ
cosθ
tanθ
O
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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
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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. 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
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Algebra 3 Assignment # 2
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
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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
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
Alg 3(11) 16 Ch 6 Trig
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.
Alg 3(11) 17 Ch 6 Trig
Algebra 3 Assignment # 4
(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
Alg 3(11) 18 Ch 6 Trig
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
Alg 3(11) 19 Ch 6 Trig
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
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