How do you find the values of the trigonometric functions of theta from the information given?
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- First of all, cotθ=1tanθ=114=4 .
- sinθ=oppositehypotenuse=−4√17.
- cscθ=1sinθ=hypotenuseopposite=−√174.
- cosθ=adjacenthypotenuse=−1√17.
- secθ=1cosθ=hypotenuseadjacent=−√17.
Answered by
0
First of all,
cot
θ
=
1
tan
θ
=
1
1
4
=
4
.
The problem says that sine is negative and we can see above that tangent is negative. Using the C-A-S-T sign rule, we determine that
θ
is in Quadrant 3.
We know that
tan
θ
=
opposite
adjacent
, so our opposite side measures
−
4
units and our adjacent side measures
−
1
unit (because
(
x
,
y
)
=
(
−
,
−
)
in quadrant 3).
We use pythagorean theorem to find the hypotenuse.
(
−
4
)
2
+
(
−
1
)
2
=
h
2
16
+
1
=
h
2
h
=
±
√
17
However, the hypotenuse can never be negative, so we only keep
√
17
.
We can now fill in all four of the other ratios:
sin
θ
=
opposite
hypotenuse
=
−
4
√
17
csc
θ
=
1
sin
θ
=
hypotenuse
opposite
=
−
√
17
4
cos
θ
=
adjacent
hypotenuse
=
−
1
√
17
sec
θ
=
1
cos
θ
=
hypotenuse
adjacent
=
−
√
17
Hopefully this helps!
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