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 30
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43You should either memorize all of these identities or learn to reconstruct them quickly by reasoning
similar to that in the discussion.
Example: Verify the identity cos (180° - θ ) = -cos θ.
(
)
θ
θ
θ
°
−
=
°
+
°
Solution:
cos
180
cos
180
cos
sin
180
sin
θ
θ
=
−
⋅
+
⋅
1
cos
0
sin
θ
=
−
cos
(
)
θ
φ
+
cos
θ
φ
=
−
Example: Verify the identity
cot
tan
.
θ
φ
sin
cos
θ
φ
Solution: We apply the identity to the quantity cos (
+
) and then simplify the fractions.
(
)
θ
φ
θ
φ
θ
φ
θ
φ
θ
φ
+
−
cos
cos
cos
sin
sin
cos
cos
sin
sin
=
=
−
θ
φ
θ
φ
θ
φ
θ
φ
sin
cos
sin
cos
sin
cos
sin
cos
θ
φ
cos
sin
θ
φ
=
−
=
−
cot
tan
θ
φ
sin
cos
Problems:
Verify each of the following identities.
θ
π
θ
θ
φ
θ
φ
θ
φ
155. cos (
+
) = -cos
156. cos (
+
) + cos (
-
) = 2 cos
cos
(
)
(
)
θ
φ
θ
φ
−
−
+
cos
cos
θ
θ
θ
φ
=
2
2
157. tan
(-
) - sec
= -1
158.
tan
tan
(
)
(
)
θ
φ
θ
φ
−
+
+
cos
cos
DOUBLE-ANGLE AND HALF-ANGLE IDENTITIES
θ
φ
The double-angle identities are obtained by setting
=
in the three sum identities:
θ
θ
θ
θ
θ
θ
θ
θ
θ
sin 2
= sin (
+
) = sin
cos
+ cos
sin
= 2 sin
cos
,
θ
θ
θ
θ
θ
θ
θ
θ
θ
2
2
cos 2
= cos (
+
) = cos
cos
- sin
sin
= cos
- sin
and
θ
θ
+
( )
(
)
tan
tan
θ
θ
θ
=
+
=
tan
2
tan
.
θ
θ
−
1
tan
tan
Thus, the double-angle identities are:
θ
θ
θ
sin 2
= 2 sin
cos
,
θ
θ
θ
2
2
cos 2
= cos
- sin
and
θ
2
tan
θ
=
tan
2
.
θ
−
2
1
tan
The half-angle identities are derived from the double-angle identities for the cosine. From the
θ
θ
2
2
Pythagorean identity cos
= 1 - sin
, so
φ
φ
φ
φ
2
2
2
cos 2
= cos
- sin
= 1 - 2 sin
.
φ
By solving for sin
, we obtain
- 30 -
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