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PTG 0016 Trigonometry and Coordinate Geometry
Tutorial 1.1: Trigonometric Functions
1) Draw each angle.
a) 135°
b) 450°
c)
2) Convert each angle to a decimal in degrees. Round the answer to two decimal places.
a) 40° 10’ 25’’
Ans: 40.17°
b) 9° 9’ 9’’
3) Convert each angle to D°M’S’’ form. Round the answer to the nearest seconds.
a) 40.32°
Ans: 40°19’12’’
b) 18.255°
4) Convert each angle in degrees to radians. Express the answer as a multiple of .
a) 30°
Ans:
b) 180°
Ans:
radians
c) -135°
5) Convert each angle in radians to degrees.
a)
Ans: 60°
b)
6) Convert each angle in degrees to radians. Express your answer in decimal form,
rounded to two decimal places.
a) -40°
Ans: -0.70 radian
b) 125°
7) Convert each angle in radians to degrees. Express your answer in decimal form,
rounded to two decimal places.
a) 3.14
Ans: 179.91°
b) 6.32
8) Find the values of the six trigonometric functions of the angle
(a)
5
12
in each figure.
(b)
3
2
(c)
1
9) Use identities to find the exact value of each of the four remaining trigonometric
functions of the acute angle .
a)
= ,
=
b)
= ,
=
10) Use the definition or identities to find the exact value of each of the remaining five
trigonometric functions of the acute angle .
a)
=
b)
=
c)
=2
11) Use Fundamental Identities and/or the Complementary Angle Theorem to find the
exact value of each expression. Do not use a calculator.
a)
°+
°
Ans: 1
b)
Ans: 1
c)
Ans: 0
d)
-
e)
f) 1 Ans: 0
g)
°-
-
°
h)
.
i)
+
= , use trigonometric identities to find the exact value of
12) Given
a)
b)
c)
d)
.
°
13) Given
=
, use trigonometric identities to find the exact value of
a)
Ans:
b)
°
Ans:
c)
Ans:
d)
e) Ans:
14) Given
a)
b)
c)
d)
=
use trigonometric identities to find the exact value of
15) Given
a)
=
use trigonometric identities to find the exact value of
Ans:
b)
Ans: 15
c)
Ans: 4
d)
Ans:
16) Given the approximation
exact value of
a)
b)
c)
d)
e)
f)
g)
h)
, use trigonometric identities to find the
17) Given f( ) =
. Find the exact value of each expression of each expression if
60 . Do not use a calculator.
a)
Ans:
=
b) f ( )
c) [f( )]2
d) 2 f( )
Ans:
e)
18) Find the exact value of each expression. Do not use a calculator.
a)
Ans:
b)
+
c)
–4
d)
°+
°
Ans:
19) Use a calculator to find the approximate value of each expression. Round the answer
to two decimal places.
a)
b)
c)
=0.47
= 0.38
= 0.31
d)
e)
= 3.73
= 0.31
20) A point on the terminal side of an angle
six trigonometric function of
a) (-3, 4)
b) (2, -3)
is given. Find the exact value of each of the
c) (
21) Use a coterminal angle to find the exact value of each expression. Do not use a
calculator.
a)
Ans:
b)
22) Name the quadrant in which the angle
a)
Ans: lies in quadrant II.
b)
23) Find the reference angle of each angle.
a) -30
Ans: reference angle = 30
b) 120
Ans: reference angle = 60
c) 210
d)
Ans: reference angle =
e)
f) -135
g)
h)
Ans: reference angle = 80
i)
24) Use the reference angle to find the exact value of each expression. Do not use a
calculator.
a)
Ans: -2
b)
Ans:
c)
d)
Ans:
e)
f)
Ans:
g)
h)
25) Find the exact value of each of the remaining trigonometric functions of
a)
=
,
Ans:
b)
=
c)
=
Ans:
,
,
d)
=
Ans:
e)
= 2,
,
26) Use the fact that the trigonometric functions are periodic to find the exact value of
each expression. Do not use a calculator.
a) sin405
Ans:
b) tan405
Ans: 1
c) csc450
Ans: 1
d) cot390
e)
f)
g)
h)
27) Use the even-odd properties to find the exact value of each expression. Do not use a
calculator.
a) sin(-60°)
Ans:
b) tan(-30°)
Ans:
c)
Ans: -1
d)
e)
28) Find the exact value of each expression. Do not use a calculator.
a)
+
Ans: -1
b)
+
29) Determine the amplitude and period of each function without graphing.
a) y = -sin( )
Ans: amplitude, A = 1
Period, T = 4π
b) y = 6 sin(πx)
c) y = -
30) Graph each function. Be sure to label key points and show at least two cycles.
a) y = 4 cos (x)
b) y = cos(4x)
Ans:
c) y = - cos(2x)
d) y = 2 sin(x) + 3
e) y = -6
+4
f) y = 5 – 3sin(2x)
Ans: