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Trigonometry chord length

karush

Well-known member
Jan 31, 2012
2,724
20190315_092610.jpg

Ok this should be just an observation solution ..
But isn't the equation for chord length
$$2r\sin{\frac{\theta}{2}}=
\textit{chord length}$$

Don't see any of the options
Derived from that..
 

Olinguito

Well-known member
Apr 22, 2018
251
Hi karush .

All the information is there. You just need to calculate the angle $\theta$ from the arc length $s=r\theta$.
 

karush

Well-known member
Jan 31, 2012
2,724

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I would start with the arc-length formula to find the subtended angle:

\(\displaystyle \theta=\frac{s}{r}=\frac{3.4}{2}=1.7\)

Then, use the law of cosines:

\(\displaystyle \overline{DF}=\sqrt{2^2+2^2-2(2)(2)\cos(1.7)}=2\sqrt{2-2\cos(1.7)}\)

Lastly, a double-angle identity for cosine:

\(\displaystyle \overline{DF}=2\sqrt{4\sin^2(0.85)}=4\sin(0.85)\)
 

karush

Well-known member
Jan 31, 2012
2,724
Ok
So that's where .85 comes from
So then it's D
 

Olinguito

Well-known member
Apr 22, 2018
251
It is asking for chord lenght!
And to do that, you need to know the angle $\theta$, don’t you? Calculate that first!
 

karush

Well-known member
Jan 31, 2012
2,724
Why of course we do!

However for this SAT question
It is only asking which
Expression to use
We should know that 1.7 is not the $\theta$ we need
and we have use sin $\theta$
So even without any calculations we should see that it is D
 
Last edited:

Olinguito

Well-known member
Apr 22, 2018
251
You were asking
Don't see any of the options
That was because you were given arc length $s=3.4$ (and radius $r=2$) but not $\theta$. I was therefore instructing you to compute $\theta$ from the formula $s=r\theta$ so you could use it in the formula $2r\sin\dfrac{\theta}2$ for the chord length.