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 27
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43θ
θ
θ
θ
θ
θ
2
2
3
2
119. (1 + tan
) cos
= 1
120. sin
= sin
- sin
cos
1
1
θ
θ
θ
θ
θ
−
=
2
122.
2
sec
121. csc
( cos
+ sin
) = cot
+ 1
θ
θ
+
−
sin
1
sin
1
θ
θ
+
1
sin
cos
1
θ
θ
+
=
=
123.
0
124.
sin
cos
θ
θ
θ
θ
−
+
cos
sin
1
cot
tan
(
)
1
θ
θ
+
=
θ
θ
θ
θ
2
125.
sec
1
sin
126. 1 = cos
sin
( tan
+ cot
)
θ
−
1
sin
1
1
θ
θ
θ
θ
−
=
+
=
127.
sec
tan
128.
csc
cot
θ
θ
θ
θ
+
−
sec
tan
csc
cot
θ
θ
+
cos
1
cos
=
θ
θ
θ
θ
6
2
4
6
129.
130. cos
= 1 - 3 sin
+ 3 sin
- sin
θ
θ
−
2
tan
sec
1
NEGATIVE-ANGLE, SUM, DIFFERENCE AND COFUNCTION IDENTITIES
θ
If
is an angle in standard position and (x, y) is a point on its terminal side, then (x, -y) is a point on the
θ
terminal side of -
. Also, (x, y) and (x, -y) are the same distance r from the origin. Consequently,
−
( )
y
θ
θ
−
=
=
−
sin
sin
,
r
( )
x
θ
θ
−
=
=
cos
cos
r
−
( )
y
θ
θ
−
=
=
−
and
tan
tan
.
x
The trigonometric identities
θ
θ
θ
θ
θ
θ
sin(-
) = -sin
, cos(-
) = cos
and tan(-
) = -tan
are called the negative-angle identities. They should be memorized.
θ
φ
The difference identity for the cosine function says that for any two angles
and
θ
φ
θ
φ
θ
φ
cos (
-
) = cos
cos
+ sin
sin
.
The derivation of this identity involves the distance formula and the Law of Cosines and is somewhat
intricate. Read it in your favorite trigonometry reference. To help you remember the difference
θ
φ
identity for the cosine, remember that when
=
this identity reduces to the Pythagorean identity
θ
θ
+
=
2
2
.
cos
sin
1
φ
φ
By replacing
by -
in the difference identity for the cosine and then using the negative angle
identities, we obtain the sum identity for the cosine:
θ
φ
θ
φ
θ
φ
cos (
+
) = cos
cos
- sin
sin
.
θ
By considering quadrant I angles (especially angles smaller than π/4 radians) one can see that if
is an
angle in standard position and (x, y) is a point on its terminal side, then (y, x) is a point on the terminal
θ
side of π/2 -
. Also, (x, y) and (y, x) are the same distance from the origin. From the definition of the
- 27 -
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