Looking for help on finding distances (quadrilateral with two sides, three angles known)

J

jakebrax

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Feb 25, 2019
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Just brushing up my trig skills. I've been staring at this problem for a day now, what am I missing.
11170
 
This is more than a "brushing-up" problem! I had to stare at it a bit before finding a likely approach.

Define angle BAC as x, and AC as d. Write two trig equations using these variables, based on triangles ABC and ADC. (When I tried this out, I didn't use actual numbers but let a=692.49, b=387.17, and A=78deg33min21sec. That let me do the work in general, without lots of long numbers.)

See what you can do to eliminate d and solve for x; from that, you should be able to find the three distances. Show us whatever work you do, and we can try to guide you along.

What contest is this from? [EDIT: I looked it up and found the problem in a sample test for a high school competition, https://cdn.ymaws.com/www.nsps.us.com/resource/resmgr/trig-star/18-19_sample.pdf.]
 
Dr.Peterson said:
This is more than a "brushing-up" problem! I had to stare at it a bit before finding a likely approach.

Define angle BAC as x, and AC as d. Write two trig equations using these variables, based on triangles ABC and ADC. (When I tried this out, I didn't use actual numbers but let a=692.49, b=387.17, and A=78deg33min21sec. That let me do the work in general, without lots of long numbers.)

See what you can do to eliminate d and solve for x; from that, you should be able to find the three distances. Show us whatever work you do, and we can try to guide you along.

What contest is this from? [EDIT: I looked it up and found the problem in a sample test for a high school competition, https://cdn.ymaws.com/www.nsps.us.com/resource/resmgr/trig-star/18-19_sample.pdf.]
Click to expand...
Dr. Peterson thank you for the response, but I am still not getting it. Basically I need to setup two equations that solve for AC or d and since AC is the same for both then I can say that the equations are equal and solve for x correct? I just cant seem to figure out the appropriate equation.
 
Please make an attempt, so I can see if you are close, or are misunderstanding my suggestion. It sounds like you have the right idea; once I see two equations, right or wrong, I'll be better able to guide you.
 
Dr.Peterson said:
Please make an attempt, so I can see if you are close, or are misunderstanding my suggestion. It sounds like you have the right idea; once I see two equations, right or wrong, I'll be better able to guide you.
Click to expand...
I was going to use the law of cosines but that seems like it has too many unknown variables so I thought. d=sinx/692.49 and d= 387.17/sin(78.56-x) so sinx/692.49=387.17/sin(78.56-x)
 
That's very close to what I got; but why is the sin on top of one and in the bottom of the other?

Fix that, then use the angle-difference formula to express everything in terms of only sin and cos of x, then solve.

I had considered using the Law of Cosines, but it didn't look like it would go anywhere.
 
Dr.Peterson said:
That's very close to what I got; but why is the sin on top of one and in the bottom of the other?

Fix that, then use the angle-difference formula to express everything in terms of only sin and cos of x, then solve.

I had considered using the Law of Cosines, but it didn't look like it would go anywhere.
Click to expand...
Wow that problem became very annoying. so....
387.17/sin(78.56-x)=692.49/sin(x)
387.17sin(x)=692.49(sin(78.56)cos(x)-cos(78.56)sin(x))
and then reduce down from there?
 
That looks about right. Now collect terms in sin(x) on one side and cos(x) on the other, and you'll be able to find tan(x).
 
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