Let θ be an angle in standard position lying in Quadrant 1. If sin(θ)= 0.227, then find the value of sin(2π−θ). Hint: Use symmetry.

Sec
θ
=
−
4
√
15
15
tan
θ
=
−
√
15
15
Explanation:
Recall that
sin
θ
=
opposite
hypotenuse
Hence, the side opposite
θ
in our question measures
1
unit and the hypotenuse measures
4
units.
Since we're dealing with right triangles, we can find the side adjacent
θ
using pythagorean theorem.
Let the adjacent side be
a
.
a
2
+
1
2
=
4
2
a
2
+
1
=
16
a
2
=
15
a
=
√
15
Now, let's define secant and tangent.
sec
θ
=
1
cos
θ
=
1
adjacent
hypotenuse
=
hypotenuse
adjacent
tan
θ
=
sin
θ
cos
θ
=
opposite
hypotenuse
adjacent
hypotenuse
=
opposite
adjacent
Applying these definitions:
sec
θ
=
4
√
15
=
4
√
15
15
tan
θ
=
1
√
15
=
√
15
15
The last thing left to do is to find the signs of these ratios. We know that we're in quadrant
I
I
, where sine is positive, and all the other ratios are negative. Since secant is related to cosine, it will be negative.
So, our final ratios are:
sec
θ
=
−
4
√
15
15
tan
θ
=
−
√
15
15
Hopefully this helps!

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