m
Find the rule of each of the following tangent functions.
b)
a)
----t---l--------
c)
(0
,X
lf x
€
10,
U,J)
'
2n], determine the
a) 4cosx:
2
:
c)
tan2x
-
.]
cos2x
- 2sinx: 2
secx
1
values of x that satisfy each of the following equations.
* cosx: 0
b)
4cos3x
d)
2cos2xf
f)
2lan2x
*
sinx-1:0
sec2x:
10
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Solve each of the following equations.
:
tanx - cotx:0
tan2x : 2secx
a)
sin2x: sinx
b) 2cos2x
d)
tanx-cotx:2
e)
s)
4sinx:
h)
3cosecx
1
c)
f)
i)
- cosx:0
cos2x sin2x : 0
4cos2x f 4cosx + 1 :0
2sin2x
GRAViTATICINAL 5["lf,iCSFl0T fFFECT When a space probe passes near a celestial body,
the gravitational pull is such that the probe deviates from its path and, in some cases,
accelerates. This effect is called the gravitational slingshot or swing-by effect.
A space probe passes close to a celestial body whose radius is 1500 km and describes an
arc of circle of 3450 km around this celestial body. A distance of 20 km separates the space
probe from the celestial body. What is the measure, in radians, of the arc of circle described
by the probe?
An t ustrat on ol Ihe t/oyager 2
space probe accelerat ng
towards Satum after passrng in
the vicinitlr of Juciter n
1
lr,'1arch
979, as a resu t of the
gravrtatona
s ngshot effect.
I
Corplete each of the following equalities.
E
rad
:
b) -r+
d) 27":
rad
e)
-f
h)
+rad:
rad
c)
75"
:
rad
f) 270":
i) -78" :
rad
rad
Determine the quadrant in which each of the following trigonometric points is located.
a)
rffi
b)'g)
c)'l#)
d)' Pl!q\
\171
e)
P(-7)
f)
s)
h)
P(s)
P(l )
P(3)
For each of the following trigonometric functions, determine:
1)
E
:
rad
30o
g) -36":
E
rad
:
a)
2) the range
the domain
3)
the period
a)
f(x)
::sinf
- 2)+4
b)
f(x):tanx*1
c)
f(x)
: -2sin 3(x + n)-5
d)
f(x):cosl(x+1)+18
e)
f(x)
: -7 cos2x +
f)
f(x):-2tan3n(x- 2)+8
(x
1
Prove each of the following identities.
a)
ETT#:1-sinx
c)
tanx +
v/,
sinx
sinx * cosx
g)
(1 + tan x)2 + (1
6t
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5
cotx:
sec xcosec
b) 1-2sin2x:2cos2x-1
d)
x
tan x
I*
-
cotx)sinxcosx
:
sin2x
-
l)- : COSX
. -.r-I COSX- :ZSeC^
| LStnx
-stnx
r\
tanx
-
(tanx
tanx)2
:
2sec2x
tun". t
h)' -lttan'x
1-t- c-ot2x
cot'x
:
sin2xsec2x
cos2x
m
Solve each of the following equations over the interval
_ tl + 6:
a)
' 3sinalx
2\
3i
-
I
.\
c) -7tan!Q
+ 5n) :
)
e)
'
[ 3tr, 3n].
. b) 8cos3(x - ri) + 4:
6
0
d) o.stinf{x_ t) +1
_\
f) 2tan2(^/ - ;J
- 2- 2
7
/ al
_\
: t;
\ 8/ VJ
2cos2lx
Solve each of the following inequalities.
a) 3r2sinL+7>10
5
b)
c) ztan|(x-1)+2<o
d) -2cosx + 2O>J,
a)
e)
' 5cos3[x
\ 4t t
f) 2tan2! - t) - 2Ji
lf
cosx:ftandxe
a)
lf
x
tanx:
a)
Xi
sin
b)
nEand
cosx
2.5
sin(x
-
0.s) + JJ <
0
*t
[0,
secx
7, determine the value of:
c) tan x
d)
e)
cosec x
cotx
xe lrr, 2r], determine the value of:
b) cotx
c)
d)
cosecx
e) secx
sin x
Determine the value of each of the following expressions.
/
a) tin(u,.tinf)
d) .or(ur..or€,
)
a-\
b)' arcsinlsinal
\ 3t
/ _\
e) arccos(cos ])
c) tan(arctan 1)
f)
/
arctan (tan
r-\
fJ
An individual is cycling at a constant speed. The height h (in cm) of the end of the valve
of the air tube in the front wheel in relation to the ground is determined by the rule
h:14sin15(f - 15) + 18 where f corresponds to the time (in s). How much time does
this wheel take to complete a full turn?
John Boyd Dunlop
(1
840-1 921) invbnted the valve-equipped air
tire in l BBB, In 1Bg1 , Edouard Michelin (1859-1940) created
the first demountable tire with an air tube.
Considering that cos a
:
1)-l]wherea €
|- l and thatsinb: Ywhere
4
be
determine the value of each of the following trigonometric expressions.
E
a)
sina
b)
tana
c)
seca
d)
coseca
e)
cota
f)
cosb
g) tan b
h)
secb
i)
cosec b
i)
k)
sin
l)
cos(a
cotb
(a + b)
-
lo,;1,
b)
Determine the sign of each of the following functions over the interval shown.
a)
b)
/_dTr I \
I ;,;l
Lt a
,l
---f---#
d)
e)
L44
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5