Download Radian Measure and Angles on the Cartesian Plane

6.2
Radian Measure and Angles
on the Cartesian Plane
GOAL
Use the Cartesian plane to evaluate the trigonometric ratios for
angles between 0 and 2p.
LEARN ABOUT the Math
Recall that the special triangles shown can be used to determine the exact
values of the primary and reciprocal trigonometric ratios for some angles
measured in degrees.
Q
B
1
A
?
45˚
2
1
P
60˚
3
2
45˚
C
1
30˚
R
How can these special triangles be used to determine the exact
values of the trigonometric ratios for angles expressed in
radians?
EXAMPLE
1
Connecting radians and the special triangles
Determine the radian measures of the angles in the special triangles, and calculate their primary trigonometric ratios.
Solution
/Q 5 60 °
/R 5 30 °
p
60 ° 5 60 ° a
b
180°
3
p
5
3
p
30 ° 5 30 ° a
b
180°
6
p
5
6
/B 5 /C 5 45 °
1
p
45 ° 5 45° a
b
180 °
4
p
5
4
/P 5 /A 5 90 °
1
p
90 ° 5 90 ° a
b
180°
2
p
5
2
1
R
p
6
1
2
Q
C
p
3
P
1
p
1 4
A
NEL
3
2
p
4
1
^PQR is the 30°, 60°, 90°
special triangle. Multiply
p
each angle by 180° to
convert from degrees to
radians.
B
^ABC is the 45°, 45°,
90° special triangle.
Multiply each angle by
p
to convert from
180°
degrees to radians.
Chapter 6
323
1.0
0
r= 2
p
4
p
4 P (1, 1)
y=1
x=1
2.0
y
p
1
5
4
"2
p
1
cos 5
4
"2
p
tan 5 1
4
sin
x
1.0
P(1, 3 )
p
"3
5
3
2
p
1
cos 5
3
2
p
tan 5 "3
3
r=2 p
1.0
6 y= 3
p
x
3
0
1.0
x=1
sin
y
1.0
0
p
1
5
6
2
p
"3
cos 5
6
2
p
1
tan 5
6
"3
sin
P( 3 , 1)
p
p
3 y=1
x
6
1.0
2.0
x= 3
r=2
csc
p
5 "2
4
p
5 "2
4
p
cot 5 1
4
sec
p
2
5
3
"3
p
sec 5 2
3
p
1
cot 5
3
"3
csc
p
52
6
p
2
sec 5
6
"3
p
cot 5 "3
6
csc
Draw each special angle
on the Cartesian plane
in standard position.
Use the trigonometric
definitions of angles on
the Cartesian plane to
determine the exact value
of each angle. Recall that
y
r
sin u 5
csc u 5
r
y
x
r
cos u 5
sec u 5
r
x
y
x
tan u 5
cot u 5
x
y
where x 2 1 y 2 5 r 2 and
r . 0.
Reflecting
324
6.2
A.
Compare the exact values of the trigonometric ratios in each special
triangle when the angles are given in radians and when the angles are
given in degrees.
B.
Explain why the strategy that is used to determine the value of a
trigonometric ratio for a given angle on the Cartesian plane is the
same when the angle is expressed in radians and when the angle is
expressed in degrees.
Radian Measure and Angles on the Cartesian Plane
NEL
6.2
APPLY the Math
EXAMPLE
2
Selecting a strategy to determine the exact
value of a trigonometric ratio
Determine the exact value of each trigonometric ratio.
p
3p
a) sin a b
b) cot a b
2
2
Solution
a)
p
is one-quarter of a full
2
y
2
P(0, 1)
1
p
2
0
–2 –1
1 2
–1
x
–2
revolution, and the point
P(0, 1) lies on the unit
circle, as shown. Draw
the angle in standard
position with its terminal
arm on the positive
y-axis. From the drawing,
x 5 0, y 5 1, and r 5 1.
y
p
sin a b 5
r
2
1
5 51
1
b)
y
3p
is three-quarters of a
2
2
3p 1
2
–2 –1 0
1 2
–1 P(0,–1)
–2
cot a
3p
x
b5
y
2
5
x
full revolution, and the
point P(0, 21) lies on
the unit circle, as shown.
Draw the angle in
standard position with its
terminal arm on the
negative y-axis. From the
drawing, x 5 0, y 5 21
and r 5 1.
0
50
21
The relationships between the principal angle, its related acute angle, and
the trigonometric ratios for angles in standard position are the same when
the angles are measured in radians and degrees.
NEL
Chapter 6
325
EXAMPLE
3
Selecting a strategy to determine the exact
value of a trigonometric ratio
Determine the exact value of each trigonometric ratio.
a) cos a 4 b
b) csc a 6 b
5p
11p
Solution A: Using the special angles
a)
y
Sketch the angle in
standard position. p is a
5p
4
half of a revolution. 4 is
halfway between p and
5p
x
3p
, and lies in the third
2
quadrant with a related
5p
p
angle of 4 2 p, or 4 .
x = –1
p
4
y = –1
y
p
is in the 1, 1, "2
4
5p
4 x
special triangle. Position
this triangle so the right
angle lies on the negative
x-axis.
r= 2
P(–1, –1)
cos a
21
5p
x
b5 5
r
4
"2
Since (21, 21) lies on
the terminal arm, x 5 21,
y 5 21, and r 5 "2.
Therefore, the cosine
ratio has a negative
value.
b)
y
11p
6
326
6.2
Sketch the angle in
11p
standard position. 6 is
x
Radian Measure and Angles on the Cartesian Plane
3p
between 2 and 2p, and
lies in the fourth quadrant
with a related angle of
11p
p
2p 2 6 , or 6 .
NEL
6.2
11p
6
y
x= 3
x
p
6 y = –1
r=2
P( 3 , –1)
csc a
r
11p
b5
y
6
5
p
is in the 1, "3, 2
6
special triangle. Position it
so that the right angle
lies on the positive x-axis.
Since the point ("3, 21)
lies on the terminal arm,
x 5 "3, y 5 21, and
r 5 2. Therefore, the csc
ratio has a negative value.
2
5 22
21
Solution B: Using a calculator
Tech
a)
Set the calculator to
radian mode. Enter the
expression.
The result is a decimal.
Entering 2
1
confirms
!2
Support
To put a graphing calculator in
radian mode, press the
MODE
key, scroll to Radian,
and press
ENTER
.
that the answer is
equivalent to this decimal.
cos a
5p
21
b5
4
"2
b)
There is no csc key on the
calculator. Use the fact
that cosecant is the
reciprocal of sine.
csc a
NEL
11p
b 5 22
6
Chapter 6
327
EXAMPLE
4
Solving a trigonometric equation that
involves radians
7
If tan u 5 2 24, where 0 # u # 2p, evaluate u to the nearest hundredth.
Solution
There are two possibilities
to consider:
x 5 24, y 5 27 and
x 5 224, y 5 7.
7
y
tan u 5 2 5
x
24
y
u
x
P(24, –7)
For the ordered pair
(24, 27), the terminal
arm of the angle u lies in
the fourth quadrant.
3p
, u , 2p
2
Use a calculator to
determine the related
acute angle by calculating
7
the inverse tan of 24.
The related angle is 0.28,
rounded to two decimal
places. Subtract 0.28
from 2p to determine
one measure of u.
2p 2 0.28 8 6.00
In the fourth quadrant, u is about 6.00.
y
P(–24, 7)
u
x
0
p 2 0.28 8 2.86
In the second quadrant, u is about 2.86.
328
6.2
Radian Measure and Angles on the Cartesian Plane
For the ordered pair
(224, 7), the terminal
arm of u lies in the second
p
quadrant, 2 , u , p, and
also has a related angle of
0.28. Subtract 0.28 from p
to determine the other
measure of u.
NEL
6.2
In Summary
Key Ideas
• The angles in the special triangles can be expressed in radians, as well as in degrees. The radian measures can be used
to determine the exact values of the trigonometric ratios for multiples of these angles between 0 and 2p.
• The strategies that are used to determine the values of the trigonometric ratios when an angle is expressed in degrees
on the Cartesian plane can also be used when the angle is expressed in radians.
The Special Triangles on the Cartesian Plane
Using a Circle of Radius 1
The Special Triangles
p
4
2
1
p
4
y
1
p
3
3
2
1
y
p
6
–1
0
P 1 , 3
2 2
1
1
p
1 4 1
p
2
4
1
2
x
1
0
–1
p
1 6 3
2
p
3
1
2
x
1
–1
–1
Need to Know
• The trigonometric ratios for any principal angle, u, in standard position can be determined by finding the related
acute angle, b, using coordinates of any point that lies on the terminal arm of the angle.
y
u
x
b
y
0
x
From the Pythagorean theorem, r 2 5 x 2 1 y 2, if r . 0.
y
x
y
sin u 5
cos u 5
tan u 5
r
r
x
r
r
x
csc u 5
sec u 5
cot u 5
y
x
y
r
P(x, y)
y
2
• The CAST rule is an easy way to remember which primary trigonometric ratios are
positive in which quadrant. Since r is always positive, the sign of each primary
ratio depends on the signs of the coordinates of the point.
• In quadrant 1, All (A) ratios are positive because both x and y are positive.
• In quadrant 2, only Sine (S) is positive, since x is negative and y is positive.
• In quadrant 3, only Tangent (T) is positive because both x and y are negative.
• In quadrant 4, only Cosine (C) is positive, since x is positive and y is negative.
NEL
1
S
T
A
0
x
C
3
4
Chapter 6
329
CHECK Your Understanding
1. For each trigonometric ratio, use a sketch to determine in which
quadrant the terminal arm of the principal angle lies, the value of
the related acute angle, and the sign of the ratio.
5p
3p
a) sin
d) sec
4
6
2p
5p
b) cos
e) cos
3
3
7p
4p
c) tan
f ) cot
3
4
2. Each of the following points lies on the terminal arm of an angle in
standard position.
i) Sketch each angle.
ii) Determine the value of r.
iii) Determine the primary trigonometric ratios for the angle.
iv) Calculate the radian value of u, to the nearest hundredth, where
0 # u # 2p.
a) (6, 8)
c) (4, 23)
b) (212, 25)
d) (0, 5)
3. Determine the primary trigonometric ratios for each angle.
p
2
7p
4
p
b) 2p
d) 2
6
4. State an equivalent expression in terms of the related acute angle.
5p
p
a) sin
c) cot a2 b
6
4
5p
7p
b) cos
d) sec
3
6
a) 2
c)
PRACTISING
5. Determine the exact value of each trigonometric ratio.
K
330
6.2
a) sin
2p
3
c) tan
11p
6
e) csc
5p
6
b) cos
5p
4
d) sin
7p
4
f ) sec
5p
3
Radian Measure and Angles on the Cartesian Plane
NEL
6.2
6. For each of the following values of cos u, determine the radian value
of u if p # u # 2p.
1
a) 2
2
b)
"3
2
c) 2
d) 2
"2
2
"3
2
e) 0
f ) 21
7. The terminal arm of an angle in standard position passes through each
of the following points. Find the radian value of the angle in the
interval 30, 2p4 , to the nearest hundredth.
a) (27, 8)
c) (3, 11)
e) (9, 10)
b) (12, 2)
d) (24, 22)
f ) (6, 21)
8. State an equivalent expression in terms of the related acute angle.
3p
4
11p
b) tan
6
a) cos
p
csc a2 b
3
2p
d) cot
3
c)
2p
6
7p
f ) sec
4
e) sin
9. A leaning flagpole, 5 m long, makes an obtuse angle with the ground.
A
If the distance from the tip of the flagpole to the ground is 3.4 m,
determine the radian measure of the obtuse angle, to the nearest
hundredth.
10. The needle of a compass makes an angle of 4 radians with the line
pointing east from the centre of the compass. The tip of the needle is
4.2 cm below the line pointing west from the centre of the compass.
How long is the needle, to the nearest hundredth of a centimetre?
11. A clock is showing the time as exactly 3:00 p.m. and 25 s. Because a
T
full minute has not passed since 3:00, the hour hand is pointing
directly at the 3 and the minute hand is pointing directly at the 12. If
the tip of the second hand is directly below the tip of the hour hand,
and if the length of the second hand is 9 cm, what is the length of the
hour hand?
p
12. If you are given an angle, u, that lies in the interval uP c , 2pd ,
2
C
how would you determine the values of the primary trigonometric
ratios for this angle?
13. You are given cos u 5 2
5
, where 0 # u # 2p.
13
a) In which quadrant(s) could the terminal arm of u lie?
b) Determine all the possible trigonometric ratios for u.
c) State all the possible radian values of u, to the nearest hundredth.
NEL
Chapter 6
331
14. Use special triangles to show that the equation
5p
cos Q 6 R 5 cos (2150°) is true.
11p
15. Show that 2 sin2 u 2 1 5 sin2 u 2 cos2 u for
.
6
16. Determine the length of AB. Find the sine, cosine, and tangent ratios
of /D, given AC 5 CD 5 8 cm.
A
B
p
6
8 cm
8 cm
C
D
17. Given that x is an acute angle, draw a diagram of both angles (in
standard position) in each of the following equalities. For each angle,
indicate the related acute angle as well as the principal angle. Then,
referring to your drawings, explain why each equality is true.
a) sin x 5 sin (p 2 x)
c) cos x 5 2cos (p 2 x)
b) sin x 5 2sin (2p 2 x)
d) tan x 5 tan (p 1 x)
Extending
18. Find the sine of the angle formed by two rays that start at the origin of
the Cartesian plane if one ray passes through the point (3!3, 3)
and the other ray passes through the point (24, 4 !3). Round your
answer to the nearest hundredth, if necessary.
19. Find the cosine of the angle formed by two rays that start at the origin
of the Cartesian plane if one ray passes through the point (6!2, 6!2)
and the other ray passes through the point (27 !3, 7). Round your
answer to the nearest hundredth, if necessary.
20. Julie noticed that the ranges of the sine and cosine functions go from
21 to 1, inclusive. She then began to wonder about the reciprocals of
these functions—that is, the cosecant and secant functions. What do
you think the ranges of these functions are? Why?
21. The terminal arm of u is in the fourth quadrant. If cot u 5 2 !3,
then calculate sin u cot u 2 cos2 u.
332
6.2
Radian Measure and Angles on the Cartesian Plane
NEL