Trigonometric Functions - Unit Circle Approach Worksheet Page 4
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1215
Objective 2:
Find the Exact Value of the Six Trigonometric Functions
of Quadrantal Angles.
Find the exact values of the six trigonometric functions of:
π
Ex. 3a
θ = 0 = 0˚
Ex. 3b
θ =
= 90˚
2
3 π
Ex. 3c
θ = π = 180˚
Ex. 3d
θ =
= 270˚
2
Solution:
a)
The terminal side of θ intersects (1, 0) on the unit circle. Thus,
y
0
uc
sin(0) = y
= 0
cos(0) = x
= 1
tan(0) =
=
= 0
uc
uc
x
1
uc
1
1
1
1
csc(0) =
=
which is undefined
sec(0) =
=
= 1
y
0
x
1
uc
uc
x
1
uc
cot(0) =
=
which is undefined
y
0
uc
b)
The terminal side of θ intersects (0, 1) on the unit circle. Thus,
π
π
sin(
) = y
= 1
cos(
) = x
= 0
uc
uc
2
2
1
1
1
π
y
π
uc
tan(
) =
=
which is undefined
csc(
) =
=
= 1
2
x
0
2
y
1
uc
uc
1
1
x
0
π
π
uc
sec(
) =
=
which is undefined
cot(
) =
=
= 0
2
x
0
2
y
1
uc
uc
c)
The terminal side of θ intersects (– 1, 0) on the unit circle. Thus,
y
0
uc
sin(π) = y
= 0
cos(π) = x
= – 1
tan(π) =
=
= 0
uc
uc
x
1
−
uc
1
1
1
1
csc(π) =
=
which is undefined
sec(π) =
=
= – 1
y
0
x
− 1
uc
uc
x
1
uc
cot(π) =
=
which is undefined
y
0
uc
d)
The terminal side of θ intersects (0, – 1) on the unit circle. Thus,
3 π
3 π
sin(
) = y
= – 1
cos(
) = x
= 0
uc
uc
2
2
3 π
1
3 π
1
1
y
uc
tan(
) =
=
which is undefined
csc(
) =
=
= – 1
2
x
0
2
y
− 1
uc
uc
3 π
1
1
3 π
x
0
uc
sec(
) =
=
which is undefined
cot(
) =
=
= 0
2
x
0
2
y
1
uc
uc
We can summarize our results:
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