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Alg 3 Ch 6.16.2 1
6.1 Angles in TRIGONOMETRY
NOTES:
I Def. of radian: 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.
r
1 rad r
1 R 5
5
2 R 5
10 ex
90° 180° 360° 270°
1
2
3
1
3
2
Alg 3 Ch 6.16.2 2
SECTORS
0 , R 2π
y
x
Alg 3 Ch 6.16.2 3
GEOMETRY REVIEW
30 – 60 – 90 ° RIGHT TRIANGLES 45 – 45 90 °
SOHCAHTOA Trig Ratios for Right Triangles
60°
30°
a 2a
3 a
45°
45°
a
a
a 2
60°
30°
1 2
3
45°
45°
1
1
2
Alg 3 Ch 6.16.2 4
Algebra 3 Assignment # 1 SKETCHPAD
Alg 3 Ch 6.16.2 5
6.2 Trigonometric Functions
Intro: The unit circle is the circle with radius = 1, center is located at the origin . I Vocabulary
A. Initial side:
B. Terminal side:
C. Coterminal angles:
The initial and terminal sides form an angle at the center
if the terminal side rotates CCW, the angle is positive if the termianl side rotates CW, the angle is negative
unit circle positive negative coterminal
30 – 60 – 90 ° RIGHT TRIANGLES 45 – 45 90 °
Right Triangle Trigonometry SOHCAHTOA
II Trig or Circular Functions θ “theta” is a variable used to represent an angle
θ
opp
adj
hyp
60
°
30
°
1 2
3
45
°
45
°
1
1
2 60
°
30
°
1 45
°
45
°
1
θ
Alg 3 Ch 6.16.2 6
SOHCAHTOA
A. sin opp hyp
θ = B. cos adj hyp
θ = C. tan opp adj
θ =
1
O
P(x,y)
A (1,0)
B (0,1)
x
y
=
=
=
sinθ
cosθ
tanθ
θ C B
A
θ C B
A
θ
A
B C
On the unit circle
θ
A
B C
10 6
C
θ
B
A
5
3
Alg 3 Ch 6.16.2 7
Triangles in the Unit Circle
I
II Reference Triangles Remember the special triangles w/ radius = 1
A. Drop ⊥ from point to xaxis.
B. Examples
1. Find the sin 3 4
π
O
Where functions are positive? Signs of Trig functions
45 30
Alg 3 Ch 6.16.2 8
2. Find the
3. Find the
4. Find the
5. Find the cos 3 4
π −
= 5 4
R π
6. Find the sin 420° =
7. Find the cos 13 6
π −
=
coterminal angles
coterminal angles
coterminal angles
π 5 sin
6
π 7 cos
4
π −
tan
3
Alg 3 Ch 6.16.2 9
8. Find sin π, cos π and tan π.
III Quadrangle Angles
Def: An angle that has its terminal side on one of the coordinate axes.
To find these angles , use the chart
A (1,0)
B (0,1)
C (1,0)
D (0,1)
1
1 sin cos
y
x
= =
= =
= =
sinθ y
cosθ x
y tanθ x
Alg 3 Ch 6.16.2 10
Complete the following tables.
Algebra 3 Assignment # 2
Radian Measure
4 3 π 7
4 π 11
6 π
π
Degree Measure 150° 90° −45° 60°
Sin
Cos
Radian Measure 2
π − 7
3 π
− 2 3 π
4 15π
Degree Measure −330° 210° 390° −270°
Sin
Cos
Alg 3 Ch 6.16.2 11
Algebra 3 Assignment # 2 Answers
Radian Measure
4 3 π 5
6 π 7
4 π
2 π 11
6 π
4 π
− π 3 π
Degree Measure 240° 150° 315° 90° 330° −45° 180° 60°
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
Radian Measure 2
π − − 11
6 π 7
3 π
− 7 6 π 2
3 π 13
6 π
4 15π 3
2 π
−
Degree Measure −90° −330° −420° 210° 120° 390° 675° −270°
Sin −1 2 1
− 2 3 −
2 1
2 3
2 1
− 2 2 1
Cos 0 2 3
2 1
− 2 3 −
2 1
2 3
2 2 0
Alg 3 Ch 6.16.2 12
6.2 Other Trigonometric Functions
A. 1
sinθ Cosecant: (csc)
1 cosθ
Secant: (sec)
1 tanθ
Cotangent: (cot)
B. Find the following values 1. csc π
4 2. sec 3π
2
3. cot − π 4
4. 17
6 cot π
C. Identities – they come from the Pythagorean Triangle
θ x
y
r=1
2 2 1 x y + =
2 2 1 cos sin θ + θ =
divide by 2 cos θ
1
x
y
divide by 2 sin θ
Alg 3 Ch 6.16.2 13
DAY 2 NOT ON THE UNIT CIRCLE 1. Find cosθ if sinθ = 2 /3 and 0 ≤ θ ≤
π 2
2. Find tanθ if sinθ = 3 /7 and π 2
≤ θ ≤ π
3. Find cscθ if cosθ = 32
− and π ≤ θ ≤ 3π 2
4. Find secθ if sinθ = 1 /3 and 3π 2
≤ 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 termninal ray.
Alg 3 Ch 6.16.2 14
Algebra 3 Assignment # 3
Complete the following tables please.
Radian Measure 3
8π 4 3π
6 π
π 5
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
Algebra 3 Assignment # 3 Answers
Radian Measure 3
8π 6 11π
4 3π
2 5π
6 π
4 3π
− π 5 3 4π
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 − 3 1
−1 3 1
1 0 3
Cot − 3 1
− 3 −1 0 3 1 3 1
Sec −2 3 2
− 2 3 2
− 2 −1 −2
Csc 3 2
−2 2 1 2 − 2 − 3 2
Radian Measure 2
3π − π 3
3 7π
− 6 5π
4 13π
6 7π
− 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 0 − 3 − 3 1
1 − 3 1
−1
Cot 0 − 3 1
− 3 1 − 3 −1 0
Sec −1 2 − 3 2
− 2 − 3 2
2
Csc 1 − 3 2
2 − 2 2 − 2 −1
Algebra 3 Review Worksheet Assignment # 4
Remaining Trig Functions
(1) Sin(θ ) = 4 5 , 0
2 π
< θ < . Find the remaining 5 trig. functions of θ .
(2) Cos(θ ) = 5 13
− , 2 π
< θ < π . Find the remaining 5 trig. functions of θ .
(3) Tan(θ ) = 2 5 , 180 270 < θ < o o . Find the remaining 5 trig. functions of θ .
(4) Sec(θ ) = 7 3 , 3 2 2 π
< θ < π . Find the remaining 5 trig. functions of θ .
(5) Csc(θ ) = 5 3
− , 3 2 π
π < θ < . Find the remaining 5 trig. functions of θ .
(6) Cot(θ ) = 3 − , 0 2 π
− < θ < . Find the remaining 5 trig. functions of θ .
(7) 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.
(8) Find the values of the six trig. functions of θ, if θ is an angle in standard position with the point ( ) 0 , 5 − on
its terminal ray.
Algebra 3 Review Worksheet Assignment # 4
Answers
(1) cos(θ ) = 3 5 , tan(θ ) = 4 3 , cot(θ ) = 3 4 , sec(θ ) = 5 3 , csc(θ ) = 5 4
(2) sin(θ ) = 1213 , tan(θ ) = − 12 5 , cot(θ ) = − 5 12 , sec(θ ) = − 13 5 , csc(θ ) = 1312
(3) sin(θ ) = 2 29
− , cos(θ ) = 5 29
− , cot(θ ) = 5 2 , sec(θ ) = 295 − , csc(θ ) = 29
2 −
(4) sin(θ ) = 2 10 7 − , cos(θ ) = 3 7 , tan(θ ) = 2 10
3 − , cot(θ ) = 3 2 10
− , csc(θ ) = 7 2 10
−
(5) sin(θ ) = 3 5 − , cos(θ ) = 4
5 − , tan(θ ) = 3 4 , cot(θ ) = 4 3 , sec(θ ) = 5 4
−
(6) sin(θ ) = 1 10
− , cos(θ ) = 3 10
, tan(θ ) = 1 3 − , sec(θ ) = 10
3 , csc(θ ) = 10 −
(7) sin(θ ) = 1213 , cos(θ ) = 5 13
− , tan(θ ) = 12 5
− , cot(θ ) = 5 12
− , sec(θ ) = 13 5
− , csc(θ ) = 1312
(8) sin(θ ) = –1 , cos(θ ) = 0 , tan(θ ) is undefined , cot(θ ) = 0 , sec(θ ) is undefined , csc(θ ) = –1
Algebra 3 Review Worksheet Assignment # 4
Algebra 3 Review Worksheet (1) Complete the following table please.
Radian Measure 3
7π 6
17π 3 8π
4 3π
−
Degree Measure 315° −90° −150° 360°
Sin
Cos
Tan
Cot
Sec
Csc
(2) Cos(θ ) = 13 5
− , θ is in Quadrant II . Find the remaining 5 trig. functions of θ .
(3) Tan(θ ) = 3 1 , θ is in Quadrant III . Find the remaining 5 trig. functions of θ .
(4) Csc(θ ) = 5 − , θ is in Quadrant IV . Find the remaining 5 trig. functions of θ .
(5) 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.
Alg 3(11) 19 Ch 6 Trig
Answers
(1) Radian Measure 3
7π 4 7π
6 17π
2 π
− 3 8π
6 5π
− 4 3π
− π 2
Degree Measure 420° 315° 510° −90° 480° −150° −135° 360°
Sin 2 3
− 2 2
2 1
−1 2 3
− 2 1
− 2 2 0
Cos 2 1
2 2
− 2 3 0 −
2 1
− 2 3
− 2 2 1
Tan 3 −1 − 3 1
− 3 3 1
1 0
Cot 3 1
−1 − 3 0 − 3 1
3 1
Sec 2 2 − 3 2
−2 − 3 2
− 2 1
Csc 3 2
− 2 2 −1 3 2
−2 − 2
(2) sin = 13 12 , tan ( ) θ = 5
12 − , cot ( ) θ = 12 5 − , sec ( ) θ = 5
13 − , csc ( ) θ = 12 13
(3) sin ( ) θ = 10 1 − , cos( ) θ =
10 3 − , cot ( ) θ = 3 , sec ( ) θ = 3
10 − , csc ( ) θ = 10 −
(4) sin ( ) θ = 5 1 − , cos( ) θ = 5
6 2 , tan ( ) θ = 6 2
1 − , cot ( ) θ = 6 2 − , sec( ) θ = 6 2
5
(5) sin(θ ) = 3 5
− , cos(θ ) = 5 4 , tan(θ ) = 3
4 − , cot(θ ) = 4
3 − , sec(θ ) = 4
5 , csc(θ ) = 5 3
−
Alg 3(11) 20 Ch 6 Trig
ADDITIONAL REVIEW
1. Convert the following to radians: a) 135˚ b) 420˚ c) 7200˚
2. Convert the following to degrees: a) 18
π b)
3
45
π c)
2
33
π
3. When angle is Θ is placed in standard position, its terminal side passes through the given point. Find the values for all six trig functions.
a) (2, 2) b) (5, 12) c) (0, 7) d) (4, 5)
4. Given the quadrant of φ and one of its six trig values, find the other five.
a) sin φ = 1
3 , φ in quadrant I b) tan φ = 2, φ in quadrant II
c) sec φ = 7
5 , tan φ < 0 d) cot φ = 1, sin φ < 0
5. Fill in the blanks for the following:
a) r = 5, s = 25π, θ = ______, A = ______
b) r = 25, s = 5π, θ = ______, A = ______
c) r = _____, s = ______ θ = 5π, A = 10π
d) r = _____, s = 2, θ = ______, A = 20
6. Find each of the following:
a) sin 4
π b) cos
3
π − c) sec ° 210
d) tan 5
4
π − e) csc 12π f) cot
7
2
π
g) cos 3630° h) sin 135 − ° i) sin 4
3
π
j) csc 5
3
π − k) cos
5
6
π l) sec 12π
7. Find cos (sin π ).
8. Find cos (sin (cot π 2 ))
Alg 3(11) 21 Ch 6 Trig
Answers:
1. 3 7
a) b) c) 40 4 3
π π π 2.
120 a) 10 b) 12 c)
11
° ° °
3. a) 2 2
sin cos tan 1 cot 1 sec 2 csc 2 2 2
θ = θ = − θ = − θ = − θ = − θ =
b) 12 5 12 5 13 13
sin cos tan cot sec csc 13 13 5 12 5 12
θ = − θ = θ = − θ = − θ = θ = −
c) sin 1 cos 0 tan und cot 0 sec und csc 1 θ = θ = θ = θ = θ = θ =
d) 5 41 4 41 41 41 5 4
sin cos tan cot sec csc 41 41 4 5 4 5
θ = θ = θ = θ = θ = θ =
4. a) 2 2 2 3 2
cos tan cot 2 2 sec csc 3 3 4 4
θ = θ = θ = θ = θ =
b) 2 5 5 5 1
sin cos cot sec 5 csc 5 5 2 2
θ = θ = − θ = − θ = − θ =
c) 5 2 6 2 6 5 6 7 6
cos sin tan cot csc 7 7 5 12 12
θ = θ = − θ = − θ = − θ = −
d) 2
cos sin tan cot 1 sec csc 2 2
θ = θ = − θ = θ = θ = θ = −
5. a) 125
5 , 2
π π b)
125 ,
5 2
π π c) r 2, s 10 = = π d)
1 r 20,
10 = θ =
6.
2 1 2 3 a) b) c) d) 1
2 2 3
3 2 e) dne f ) 0 g) h)
2 2
3 2 3 3 i) j) k) l) 1
2 3 2
− −
−
− −
Alg 3(11) 22 Ch 6 Trig 7. 1 8. 1
Alg 3(11) 23 Ch 6 Trig Extra Review –
1. Convert the following to radians a) 7200˚ b) 300˚ c) .2
2. Convert the following to degrees:
a) π 5
4 b)
π 180
c) 1 radian
3. When angle is Θ is placed in standard position, its terminal side passes through the point (3, 5). Find the values for all six trig functions.
4. When angle is Θ is placed in standard position, its terminal side passes through the point (6, 8). Find the values for all six trig functions.
5. Given that cos φ = 24
25 ,
π − 3
2 < φ< 2π. Find the value of the other 5 trig functions.
6. Given that cot φ = 5 and csc φ <0. Find the value of the other 5 trig functions.
7. If the arc length of a circle is 30 cm and the area of the sector it intercepts is 180 cm 2 , find the radius of the circle and the angle of the sector.
8. Find each of the following
a) sin ° 225 b) cos π
− 6
c) sec π 7
2 d) tan ° 1800 e) csc
π 2
3
f) cot π 7 g) sin ° 210 h) sin π
− 2
3 i) cos 0° k) sec
π − 11
4
Alg 3(11) 24 Ch 6 Trig
ANSWERS
1. a) π 40 b) π
− 5
3 c)
π 900
2. a) 225° b) 1° c) °
π 180
3. = = − = − = − = − = 5 34 3 34 34 34 5 3
sin cos tan cot sec csc 34 34 3 5 3 5
4. = = = = = = 4 3 4 3 5 5
sin cos tan cot sec csc 5 5 3 4 3 4
5. = = = = = 7 7 24 25 25
sin tan cot sec csc 25 24 7 24 7
6. = − = − = = − 26 5 26 1 26
sin cos = tan sec csc 26 26 26 5 5
7. Radius: 12 Angle: 5/2
8. a) − 2
2 b)
3
2 c) dne d) 0 e)
2 3
3
9. f) dne g) − 1
2 h) −
3
2 i) 1 k) − 2
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