Answer:
Find sin 2x, knowing tan x = -2, and cos x > 0
Ans:
sin
2
x
=
4
5
Explanation:
3 Trig identities to be used:
1
+
tan
2
x
=
1
cos
2
x
(1)
sin
2
x
+
cos
2
x
=
1
(2)
sin
2
x
=
2
sin
x
.
cos
x
(3)
Given tan x = -2. First find cos x and sin x
(1) -->
1
+
4
=
1
cos
2
x
-->
cos
2
x
=
1
5
-->
cos
x
=
±
1
√
5
.
Since cos x > 0, then
cos
x
=
1
√
5
.
(2) -->
sin
2
x
=
1
−
cos
2
x
=
1
−
1
5
=
4
5
-->
sin
x
=
±
2
√
5
.
Since cos x > 0 then
sin
x
=
2
√
5
.
(3) -->
sin
2
x
=
2
sin
x
.
cos
x
=
2
(
1
√
5
)
(
2
√
5
)
=
4
5
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