Question

If sintheta=2/3 which of the following are possible?
sectheta= -3/2 and tantheta= 2/root5
costheta= -root5/3 and tantheta= 2/root5
costheta= root5/3 and tantheta= 2/root5
sectheta= 3/root5 and tantheta= 2/root5

Answer 1

First find the missing side. We know that sin\(\theta\)=\(\frac{2}{3}\), so the opposite side is \(2\), and the hypotenuse is \(3\). Now we need to find the adjacent.

Use the Pythagorean theorem.
\(c^2=a^2+b^2\)
\(3^2=2^2+b^2\)
\(b^2=9-4\)
\(b=\sqrt{5}\)

So the adjacent side is \(\sqrt{5}\).

Answer 2

okay im with it so far

Answer 3

So now we know the three sides:

opposite: 2
adjacent: \(\sqrt {5}\)
hypotenuse: 3

----
We know that
\(\huge cos\theta\)=\(\huge \frac {adjacent}{hypotenuse}\)


so

\(\huge cos\theta\)=\(\huge \frac {\sqrt{5} }{3}\)

Answer 4

Now that we have the sides which are
2,3, and -root5
we know that
sin=-3/root5
cos=2/5
tan=3/2
sec=-root5/2
csc= -root5/3
and cot= 2/3

Answer 5

i didnt mean negative root 5 just root 5

Answer 6

but i also see that i messed up

Answer 7

Oops okay i had them in the wrong order then

Answer 8

And \(\tan\theta =\huge \frac {oposite}{adjacent}\), so \(\tan\theta =\huge \frac {2}{\sqrt{5}}\) or \(\huge \frac {2\sqrt{5}} {5}\) .

Answer 9

opposite*

Answer 10

So that means that the last two in my question are the best two options

Answer 11

Yes! :)