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 31
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43φ
−
1
cos
2
φ
=
±
sin
.
2
θ
φ
Finally, by replacing
with
, we obtain
2
θ
θ
−
1
cos
=
±
sin
,
2
2
θ
where the sign is determined from the quadrant of
. By a similar calculation
2
θ
θ
+
1
cos
=
±
cos
,
2
2
θ
where the sign is determined by the quadrant of
.
2
θ
Finally, to obtain a formula for tan
, use the half-angle identities for the sine and cosine in
2
( )
θ
θ
sin
2
=
( )
tan
θ
2
cos
2
and get
θ
θ
−
1
cos
=
±
tan
,
θ
+
2
1
cos
θ
where, again, the sign is determined by the quadrant of
.
2
You should memorize the double-angle and half-angle identities by learning to reconstruct them
quickly from previous identities as in the discussion above.
Problems:
θ
θ
φ
θ
φ
159. sin 2
= __________________________
160. _____________ = sin
cos
- cos
sin
θ
θ
φ
2
161. 1 - 2 sin
= _______________________
162. tan (
+
) = _______________________
θ
+
1
cos
θ
φ
±
164. _______________________ = sin (
+
)
163. _____________________ =
2
θ
−
1
cos
θ
±
=
165. cos 2
= ___________________________
166.
______________________
θ
+
1
cos
θ
θ
=
2
167. _______________________ = 2 cos
- 1
168.
____________________________
sin
2
θ
φ
−
tan
tan
θ
φ
=
170. sin (
-
) = _______________________
169. ____________________
θ
φ
+
1
tan
tan
θ
θ
θ
2
2
171. tan 2
= __________________________
172. _____________________ = cos
- sin
θ
θ
θ
φ
θ
φ
173. 2 sin
cos
= ______________________
174. sin
cos
+ cos
sin
= _____________
- 31 -
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