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Lesson 6.1 Exercises, pages 474–480
A
3. Use technology to determine the value of each trigonometric ratio
to the nearest thousandth.
a) sin 415°
b) cos (⫺65°)
⬟ 0.819
c) cot 72°
⬟ 0.423
ⴝ
d) csc 285°
1
tan 72°
ⴝ
⬟ 0.325
1
sin 285°
⬟ ⴚ1.035
4. Sketch each angle in standard position, then identify the reference
angle.
b) ⫺350°
a) 460°
y
y
460°
x
⫺350°
x
O
A coterminal angle is:
460° ⴚ 360° ⴝ 100°
The reference angle is:
180° ⴚ 100° ⴝ 80°
©P DO NOT COPY.
A coterminal angle is:
360° ⴚ 350° ⴝ 10°
This is also the reference angle.
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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d) ⫺500°
c) 695°
y
y
x
695°
A coterminal angle is:
ⴚ720° ⴙ 695° ⴝ ⴚ25°
The reference angle is: 25°
x
⫺500°
A coterminal angle is:
360° ⴚ 500° ⴝ ⴚ140°
The reference angle is:
180° ⴚ 140° ⴝ 40°
5. For each angle in standard position below:
i) Determine the measures of angles that are coterminal with the
angle in the given domain.
ii) Write an expression for the measures of all the angles that are
coterminal with the angle in standard position.
a) 75°;
for ⫺500° ≤ u ≤ 500°
i) Between 500° and 0°, the
coterminal angles are:
75°; and 75° ⴙ 360° ⴝ 435°
Between 0° and ⴚ500°, the
coterminal angle is:
75° ⴚ 360° ⴝ ⴚ285°
ii) The measures of all the
angles coterminal with 75°
can be represented by the
expression:
75° ⴙ k360°, k ç ⺪
c) 215°;
for ⫺700° ≤ u ≤ 700°
i) Between 700° and 0°, the
coterminal angles are:
215°; and
215° ⴙ 360° ⴝ 575°
Between 0° and ⴚ700°, the
coterminal angles are:
215° ⴚ 360° ⴝ ⴚ145°; and
215° ⴚ 2(360°) ⴝ ⴚ505°
ii) The measures of all the
angles coterminal with 215°
can be represented by the
expression: 215° ⴙ k360°,
kç⺪
2
b) ⫺105°;
for ⫺600° ≤ u ≤ 600°
i) Between 600° and 0°, the
coterminal angle is:
ⴚ105° ⴙ 360° ⴝ 255°
Between 0° and ⴚ600°, the
coterminal angles are:
ⴚ105°; and
ⴚ105° ⴚ 360° ⴝ ⴚ465°
ii) The measures of all the angles
coterminal with ⴚ105° can be
represented by the expression:
ⴚ105° ⴙ k360°, k ç ⺪
d) ⫺290°;
for ⫺800° ≤ u ≤ 800°
i) Between 800° and 0°, the
coterminal angles are:
ⴚ290° ⴙ 360° ⴝ 70°;
ⴚ290° ⴙ 2(360°) ⴝ 430°; and
ⴚ290° ⴙ 3(360°) ⴝ 790°
Between 0° and ⴚ800°, the
coterminal angles are:
ⴚ290°; and
ⴚ290° ⴚ 360° ⴝ ⴚ650°
ii) The measures of all the angles
coterminal with ⴚ290° can be
represented by the expression:
ⴚ290° ⴙ k360°, k ç ⺪
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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B
6. Use exact values to complete this table.
U
sin U
cos U
tan U
csc U
sec U
cot U
0°, 0
0
1
0
–
1
–
1
√
3
2
2
√
3
√
30°,
␲
6
1
2
45°,
␲
4
1
√
2
1
√
2
␲
60°,
3
√
3
2
1
2
1
0
ⴚ
90°,
␲
2
3
2
120°,
2␲
3
√
3
2
135°,
3␲
4
1
√
2
3
–
1
2
1
2
ⴚ√
1
2
225°,
5␲
4
ⴚ√
240°,
4␲
3
ⴚ
270°,
3␲
2
ⴚ1
1
2
3
2
0
2
√
3
ⴚ2
ⴚ√
√
2
–
2
–
3
2
1
√
3
ⴚ2
ⴚ√
1
2
1
√
ⴚ 2
√
ⴚ 2
1
ⴚ√
ⴚ2
1
√
3
–
0
ⴚ√
1
2
√
3
2
3
2
3
2
3
3
2
1
2
√
ⴚ 3
ⴚ√
2
1
2
1
√
2
ⴚ1
√
ⴚ 2
√
ⴚ√
ⴚ2
2
√
3
0
–
1
1
ⴚ
2
360°, 2␲
0
√
3
2
1
ⴚ1
ⴚ1
1
3
ⴚ1
11␲
330°,
6
1
3
√
–
ⴚ√
©P DO NOT COPY.
–
0
√
ⴚ
ⴚ
1
–
ⴚ√
√
1
√
3
0
√
ⴚ
2
√
ⴚ 3
ⴚ1
ⴚ
2
√
3
1
2
ⴚ√
0
7␲
6
√
2
180°, ␲
210°,
3
ⴚ√
3
2
ⴚ
7␲
4
ⴚ1
√
1
2
315°,
√
ⴚ 3
√
√
2
√
5␲
150°,
6
5␲
300°,
3
1
1
3
2
3
You will return to this table when
you complete Lesson 6.3
Exercises.
√
3
1
3
2
ⴚ1
√
ⴚ 3
–
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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7. Determine the exact values of the 6 trigonometric ratios for each
angle.
a) 480°
A coterminal angle is: 480º ⴚ 360º ⴝ 120º
From the completed table in question 6:
sin 480º ⴝ √
sin 120º
cos 480º ⴝ cos 120º
3
ⴝ
2
tan 480º ⴝ tan 120º
√
ⴝⴚ 3
1
ⴝⴚ
2
1
sin 120°
2
ⴝ√
3
csc 480º ⴝ
sec 480º ⴝ
1
tan 120°
1
ⴝ ⴚ√
3
1
cos 120°
cot 480º ⴝ
ⴝ ⴚ2
b) ⫺855°
A coterminal angle is: 1080º ⴚ 855º ⴝ 225º
From the completed table in question 6:
sin (ⴚ855º) ⴝ sin 225º
cos (ⴚ855º) ⴝ cos 225º
csc (ⴚ855º) ⴝ
tan (ⴚ855º) ⴝ tan 225º
1
ⴝ ⴚ√
2
1
ⴝ ⴚ√
2
1
sin 225°
sec (ⴚ855º) ⴝ
√
ⴝⴚ 2
ⴝ1
1
cos 225°
√
ⴝⴚ 2
cot (ⴚ855º) ⴝ
1
tan 225°
ⴝ1
8. For each point P(x, y) on the terminal arm of an angle u in standard
position, determine the exact values of the six trigonometric ratios
for u.
b) P(3, ⫺4)
a) P(2, 1)
Let the distance between the
origin and P be r.
Use: x2 ⴙ y2 ⴝ r2
Substitute: x ⴝ 2, y ⴝ 1
22 ⴙ 12 ⴝ r2
√
rⴝ 5
The terminal arm lies in
Quadrant 1.
√
1
sin U ⴝ √
csc U ⴝ 5
5
2
cos U ⴝ √
5
1
tan U ⴝ
2
4
√
sec U ⴝ
5
2
cot U ⴝ 2
Let the distance between the
origin and P be r.
Use: x2 ⴙ y2 ⴝ r2
Substitute: x ⴝ 3, y ⴝ ⴚ4
32 ⴙ (ⴚ4) 2 ⴝ r2
rⴝ5
The terminal arm lies in
Quadrant 4.
sin U ⴝ ⴚ
cos U ⴝ
4
5
3
5
tan U ⴝ ⴚ
csc U ⴝ ⴚ
sec U ⴝ
4
3
5
4
5
3
cot U ⴝ ⴚ
3
4
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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9. For each point P(x, y) on the terminal arm of an angle u in standard
position, determine possible measures of u in the given domain.
Give the answers to the nearest degree.
b) P(⫺4, ⫺8);
for ⫺360° ≤ u ≤ 360°
a) P(⫺4, 2);
for ⫺360° ≤ u ≤ 0°
The terminal arm of angle U lies
in Quadrant 2.
The reference angle is:
The terminal arm of angle U lies
in Quadrant 3.
The reference angle is:
tanⴚ1 a b ⬟ 27°
tanⴚ1 a b ⬟ 63°
So, U ⬟ 180° ⴚ 27°, or 153°
The angle between ⴚ360° and
0° that is coterminal with 153° is:
ⴚ360° ⴙ 153° ⴝ ⴚ207°
Possible value of U is:
approximately ⴚ207°
So, U ⬟ 180° ⴙ 63°, or 243°
The angle between ⴚ360° and 0°
that is coterminal with 243° is:
ⴚ360° ⴙ 243° ⴝ ⴚ117°
Possible values of U are
approximately: 243° and ⴚ117°
2
4
8
4
10. For each trigonometric ratio, determine the exact values of the
other 5 trigonometric ratios for u in the given domain.
√
1
a) cos u ⫽ √ ;
b) cot u ⫽ ⫺ 3;
2
for 0° ≤ u ≤ 180°
for 90° ≤ u ≤ 270°
Use the completed table in question 6.
From the given domain, U ⴝ 45°
√
1
Then: cos U ⴝ √ , sec U ⴝ 2 ,
2
√
1
sin U ⴝ √ , csc U ⴝ 2,
2
tan U ⴝ 1, and cot U ⴝ 1
√
cot U ⴝ ⴚ 3 for U ⴝ 150° or U ⴝ 330°
From the given domain, U ⴝ 150°
√
1
Then: cot U ⴝ ⴚ 3, tan U ⴝ ⴚ√ ,
1
2
3
sin U ⴝ , csc U ⴝ 2,
√
cos U ⴝ ⴚ
3
2
, and sec U ⴝ ⴚ√
2
3
11. For each value of the trigonometric ratio below, determine possible
measures of angle u in the given domain. Give the angles to the
nearest degree.
1
a) sin u ⫽ ⫺2 ; for ⫺360° ≤ u ≤ 360°
Since sin U is negative, the terminal arm of angle U lies in Quadrant 3 or 4.
For the domain 0° ◊ U ◊ 360°, from the completed table in question 6,
U ⴝ 210º or U ⴝ 330º
For the domain ⴚ360° ◊ U ◊ 0°, U ⴝ ⴚ150º or U ⴝ ⴚ30º
The angles are: ⴚ150°, ⴚ30°, 210°, and 330°
©P DO NOT COPY.
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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b) cot u ⫽ 1; for 0° ≤ u ≤ 720°
Since cot U is positive, the terminal arm of angle U lies in Quadrant 1 or 3.
For the domain 0° ◊ U ◊ 360°, from the completed table in question 6,
U ⴝ 45º or U ⴝ 225º
For the domain 360° ◊ U ◊ 720°,
U ⴝ 45º ⴙ 360°
or
U ⴝ 225° ⴙ 360°
ⴝ 405°
ⴝ 585°
The angles are: 45°, 225°, 405°, and 585°
c) sec u ⫽ ⫺11; for 0° ≤ u ≤ 360°
Since sec U is negative, the terminal arm of angle U lies in Quadrant 2 or 3.
The reference angle is: cosⴚ1 a b ⬟ 85º
1
11
For the domain 0° ◊ U ◊ 360°:
In Quadrant 2, U ⬟ 180º ⴚ 85º, or approximately 95º
In Quadrant 3, U ⬟ 180º ⴙ 85º, or approximately 265º
To the nearest degree, the angles are: 95º and 265º
C
12. Perpendicular lines are drawn from the
axes to the terminal arm of an angle u
in standard position. Lines DC and EF
are tangents to the unit circle. Explain
why each trigonometric ratio is equal
to the length of the indicated line
segment.
a) PA ⫽ sin u
In right ^ OPA,
PA
OP
PA
sin U ⴝ
1
sin U ⴝ
So, PA ⴝ sin U
y
E
F
D
P
B
1
␪
O
x
A
C
b) PB ⫽ cos u
In right ^ BOP,
∠BOP ⴝ 90º ⴚ U
So, ∠OPB ⴝ U
PB
OP
PB
cos U ⴝ
1
cos U ⴝ
So, PB ⴝ cos U
c) OF ⫽ csc u
In right ^ OEF,
∠EFO ⴝ U
EO
sin U ⴝ
OF
1
sin U ⴝ
OF
d) DC ⫽ tan u
In right ^ DOC,
DC
OC
DC
tan U ⴝ
1
tan U ⴝ
So, DC ⴝ tan U
So, OF ⴝ csc U
6
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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e) FE ⫽ cot u
In right ^ OEF,
∠EFO ⴝ U
OE
tan U ⴝ
EF
1
tan U ⴝ
EF
Quit
f) DO ⫽ sec u
In right ^ DOC,
OC
DO
1
cos U ⴝ
DO
cos U ⴝ
So, DO ⴝ sec U
So, EF ⴝ cot U
©P DO NOT COPY.
6.1 Trigonometric Ratios for Any Angle in Standard Position—Solutions
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