Trigonometry

[kris.yarmak123]

The triangle PQR has Q = 43 degrees, p=20cm and q=15cm. The smallest value of P+R+r, correct to the nearest tenth is ____.
Please help me

 

[MarkFL]

We can't very well add two angles and a side length, since they are not dimensionally equivalent.

 

[Dr.Peterson]

Have you sketched the triangle? You should recognize that this is the SSA case, which may have two solutions; that explains the "smallest value" part of the problem.

Solve the triangle as you have been taught (using the Law of Sines, probably), and calculate P+R+r for each solution (if there is at least one solution), then choose the smallest value.

If you need more help, please show your work, or tell us where you have trouble solving the triangle.

Presumably, what they mean by P+R+r is "the sum of the degree values of angles P and R, and side r". This is mathematically meaningless, but is a way to make it harder to guess an answer on a multiple-choice problem.

 

[kris.yarmak123]

Well it’s numerical response and I’m bad at most of them. So I got angle P = 65.4 degrees as the smallest value, angle R = 71.6 degrees and r=20.9.

 

[Dr.Peterson]

So I got angle P = 65.4 degrees as the smallest value, angle R = 71.6 degrees and r=20.9.

This is one of the two triangles possible, as I mentioned; and not the one they want. Good work so far; but don't stop here!

It is easy to miss the second solution if you are not aware of it; this is why we explicitly each about the "ambiguous case" of SSA, and why I mentioned it from the start.

To be more specific: There are two angles with a given sine: the inverse sine, and its supplement. So what is the other possibility for P? What do you get for R and r?

 

[kris.yarmak123]

The other possibility for P is 114.6°. for R I guess it's 22.4°. and r = 20.9 or 8.4

 

[Dr.Peterson]

For this case, r=8.4.

Now add up P+R+r for each case, pick the smallest, and you have the answer.

 

[kris.yarmak123]

For this case, r=8.4.

Now add up P+R+r for each case, pick the smallest, and you have the answer.

So the answer is 96.2 right?

 

[Dr.Peterson]

How did you get that? Please show the additions you did for each case.

 

[kris.yarmak123]

How did you get that? Please show the additions you did for each case.

Well I added 65.4°+22.4°+8.4. I'm a little confused

 

[Dr.Peterson]

Well I added 65.4°+22.4°+8.4. I'm a little confused

There are two cases; you are mixing them together. It asked for "the smallest value of P+R+r", not the sum of the smallest value of each!

Here are the two cases, as you stated them:

So I got angle P = 65.4 degrees as the smallest value, angle R = 71.6 degrees and r=20.9.

The other possibility for P is 114.6°. for R I guess it's 22.4°. and r = 20.9 or 8.4

Add each triple of numbers separately.

 

[kris.yarmak123]

Oh got it. So I added 65.4°+71.6°+20.9 and got 157.9. and for the other one I added 114.6°+22.4°+8.4 and got 145.4
. So therefore P+R+r = 145.4 right?