Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 4
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43θ
θ
θ
θ
θ
θ
Notice that cot
, sec
, and csc
are respectively the reciprocals of tan
, cos
and sin
, and that the
fundamental identity
θ
θ
2
2
sin
+ cos
= 1
is just another way of expressing the Pythagorean Theorem.
By examining the various cases, we can see that the sine function
is positive for angles whose terminal side is in the first or second
quadrant and negative for angles in quadrants III and IV. Similarly,
the cosine is positive in quadrants I and IV and negative in
quadrants II and III; while the tangent is positive in quadrants I
and III and negative in quadrants II and IV.
Figure T8
5
φ
φ
φ
−
Example: The angle
is a quadrant II angle and tan
=
. What is the value of cos
?
8
φ
Solution: Draw a figure so that
has its terminal side
5
5
φ
− .
in quadrant II and so that tan
=
=
−
8
8
φ
Then
=
+
=
+
−
=
2
2
2
2
r
x
y
(
) 5
(
) 8
89
.
Hence,
−
−
8
8
89
φ
=
=
Figure T9
cos
.
89
89
Example: Find all angles θ between 0 and 2π such that tan θ = -2.7933.
θ
θ
Solution: We need to determine the quadrants of the two angles
and
between 0 and 2π such
1
2
that tan θ = -2.7933. Since tan θ = -2.7933 is negative and the tangent is negative in
θ
θ
quadrants II and IV,
is in quadrant II and
is in quadrant IV.
1
2
θ
θ
Next we need to find the reference angle for
and
.
1
2
θ
= ′
−
=
tan
. 2
7933
. 2
7933
θ
′ =1.2270
θ
θ
Make a sketch of the angles
and
in the correct
1
2
quadrants with reference angle θ ′ =1.2270 as in
Figure T10.
θ
θ
Determine
and
using Figure T10 as a guide:
1
2
θ
π
θ
= ′
π
=
−
−
=
. 1
2270
. 1
9146
1
and
θ
π
θ
= ′
π
=
−
−
=
2
2
. 1
2270
. 5
0562
Figure T10
2
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