Find a sine

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marinaa11

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Hi! I am struggling with this task: " find the sine of the angle that forms the sidewall of this pyramid with the base plane ". This pyramid is a
tetrahedron and it's edge equals to 8.

1610187372534.png

As I understood, I have to find the sine of the angle SBC or am I mistaken? Moreover, to find a sine shouldn't it be a right triangle? I am sorry, I really do not have any concrete ideas. Could you, please give me any hints? Thank you.
 
marinaa11 said:
Hi! I am struggling with this task: " find the sine of the angle that forms the sidewall of this pyramid with the base plane ". This pyramid is a
tetrahedron and it's edge equals to 8.

View attachment 24248

As I understood, I have to find the sine of the angle SBC or am I mistaken? Moreover, to find a sine shouldn't it be a right triangle? I am sorry, I really do not have any concrete ideas. Could you, please give me any hints? Thank you.
Click to expand...
No, SBC is an angle between 2 line segments. The angle you need is between 2 planes. Read the problem more carefully.
It's easier to find sine of an angle on a right triangle - you can use side length ratios. So you need to construct it. Look up the definition of an angle between planes. Then construct a triangle with this angle but in such a way that you can calculate its side lengths.
 
marinaa11 said:
Hi! I am struggling with this task: " find the sine of the angle that forms the sidewall of this pyramid with the base plane ". This pyramid is a
tetrahedron and it's edge equals to 8.

View attachment 24248

As I understood, I have to find the sine of the angle SBC or am I mistaken? Moreover, to find a sine shouldn't it be a right triangle? I am sorry, I really do not have any concrete ideas. Could you, please give me any hints? Thank you.
Click to expand...
The problem could be worded better. I assume they mean a regular tetrahedron, and the angle is between a lateral face such as SBC and the base, ABC.

You might consider the altitude from S to plane ABC. This will require some knowledge of solid geometry.
 
There are two misconceptions in your question.

First, you are asked to find the angle between two planes, not two lines. What do you know about the angle between planes?

Second, you are usually first given a definition of sine in the context of a plane right triangle. But the sine is also a function that relates two numbers, and that function can be and is used in contexts other than that of plane right triangles. For example, there is a law of sines that relates to the angles and sides of any type of plane triangle. In a geometric context, using the sine frequently requires first constructing right triangles that relate to the problem at hand.
 
JeffM said:
Second, you are usually first given a definition of sine in the context of a plane right triangle. But the sine is also a function that relates two numbers, and that function can be and is used in contexts other than that of plane right triangles. For example, there is a law of sines that relates to the angles and sides of any type of plane triangle. In a geometric context, using the sine frequently requires first constructing right triangles that relate to the problem at hand.
Click to expand...
This made me realize that there is a nice way to solve the problem without using a right triangle at all (well, apart from the Pythagorean theorem), which will require less geometry.

Think about a "vertical" plane through S and A, perpendicular to ABC and passing through a point X on BC. The angle you are asked for is SXA, and you can easily find the lengths of the sides of that triangle. Then use one of those laws you've learned for oblique triangles.

But all this depends on fully understanding what the angle between two planes is.
 
Thanks a lot for your response and explanations! Therefore, I managed to complete the second problem of this task - it required to find the cosine between the side edge (sorry if it's not the right definition) and the base plane. If drew a sketch so I would be easier to understand in Math's language :) So, I find that cosine, but I had answers and the cosine should be three roots out of three, not one third...
And going back to finding the sine between the sidewall of this pyramid with the base plane - I still do not get which exactly angle it is: as I understand, the sidewall is the triangle SAC and the plane base is ABC, so should I write more letters on my sketch to define that sine?

1610267782243.png
 
marinaa11 said:
Thanks a lot for your response and explanations! Therefore, I managed to complete the second problem of this task - it required to find the cosine between the side edge (sorry if it's not the right definition) and the base plane. If drew a sketch so I would be easier to understand in Math's language :) So, I find that cosine, but I had answers and the cosine should be three roots out of three, not one third...
And going back to finding the sine between the sidewall of this pyramid with the base plane - I still do not get which exactly angle it is: as I understand, the sidewall is the triangle SAC and the plane base is ABC, so should I write more letters on my sketch to define that sine?

View attachment 24262
Click to expand...
Where did the first 1/3 come from?
 
A question does not necessarily mean you are wrong. You are being asked to explain your reasoning. Of course, frequently you are asked for such an explanation because your reasoning is indeed wrong, but if only to see why it was wrong, you will learn more by trying to trace out your reasoning step by step.

And if you mean by “three roots out of three” [MATH]3 \sqrt{3} > 1[/MATH], that definitely is an error because the cosine never exceeds 1.
 
Well, when the problem is that I have to find OE or SO but I am not sure about what exactly rule to use as I know only SE and the fact that triangle ABC is equilateral
 
marinaa11 said:
Do you mean when I calculated OE? I thought that OE is one-third of AE. I guess I am mistaken...
Click to expand...
Actually, you are correct; and I would have hoped you would answer by stating the theorem that supports it.

Was this just a guess, or were you thinking of the fact that O is the intersection of the medians of ABC?
 
Note - the problem asked you to find the sine, whereas you calculated the cosine.
 
Dr.Peterson said:
Actually, you are correct; and I would have hoped you would answer by stating the theorem that supports it.

Was this just a guess, or were you thinking of the fact that O is the intersection of the medians of ABC?
Click to expand...
Honestly, it was my premonition but I kept in mind the theorem of the intersection of the medians (I was not sure using it here). The problem is that my answer does not coincide with the answer which is in the book and I really want to figure it out. The answer in the book ( angle's SEO cosine ) is three roots out of three and as JeffM noticed, the cosine never exceeds 1 ... Maybe it is the problem in my calculated OE.
And yes, this task has 2 parts: one is to find the sine of the angle that forms the sidewall of this pyramid with the base plane and I have not done it yet as I do not understand which angle it would be... and the second part is to find the cosine between the side edge and the base plane. I thought at least I managed to deal with the second part but, unfortunately, the answers do not coincide :(
 
marinaa11 said:
Honestly, it was my premonition but I kept in mind the theorem of the intersection of the medians (I was not sure using it here). The problem is that my answer does not coincide with the answer which is in the book and I really want to figure it out. The answer in the book ( angle's SEO cosine ) is three roots out of three and as JeffM noticed, the cosine never exceeds 1 ... Maybe it is the problem in my calculated OE.
And yes, this task has 2 parts: one is to find the sine of the angle that forms the sidewall of this pyramid with the base plane and I have not done it yet as I do not understand which angle it would be... and the second part is to find the cosine between the side edge and the base plane. I thought at least I managed to deal with the second part but, unfortunately, the answers do not coincide :(
Click to expand...
If your problem is a mismatch with the book's answer, why didn't you say that from the start? Maybe they are wrong, or you are just not seeing that they are equal. This is why we ask you to tell us where you are stuck, to avoid wasting time and get to the heart of the matter!

The phrase "three roots out of three" is not meaningful; are you saying JeffM was right that it says three times the square root of three (\(3\sqrt{3}\))? Then clearly the book is wrong.

And your work and answer are correct, as far as they go.

Please show us the entire problem; an image will be good, even if it is not in English. (If you can also copy and paste the answer in text form so we can more easily try to translate it, that would be good.) I'd like to see their complete answer, too.
 
I just was insecure about my ways of solving this problem I am really thankful for all responds and I gained some new information.
b.jpg
It is all about the number 25: regular triangular pyramid's SABC all edges are equal to 8 cm (it is a tetrahedron).
1. Find the surface area of this pyramid.
2. Prove, that pyramid's volume is equal to...
3. Find the sine of the angle that forms the sidewall of this pyramid with the base plane.
4. Find the cosine between the side edge and the base plane.
5. Find the distance from SA and BC.
I am stuck in 25.3 and 25.4
These are mine solutions:
1610302863345.png

These are answers form the book:
1610302969144.png
 
marinaa11 said:
I just was insecure about my ways of solving this problem I am really thankful for all responds and I gained some new information.
View attachment 24270
It is all about the number 25: regular triangular pyramid's SABC all edges are equal to 8 cm (it is a tetrahedron).
1. Find the surface area of this pyramid.
2. Prove, that pyramid's volume is equal to...
3. Find the sine of the angle that forms the sidewall of this pyramid with the base plane.
4. Find the cosine between the side edge and the base plane.
5. Find the distance from SA and BC.
I am stuck in 25.3 and 25.4
These are mine solutions:
View attachment 24271

These are answers form the book:
View attachment 24272
Click to expand...
25.3 As I wrote above you need sine, not cosine. But if you know how to calculate cosine, it should be easy to find the sine - just use the correct side length in the ratio.
25.4 This is a different angle. You can use a similar approach - construct a right triangle that contains the angle in question, calculate its side lengths, calculate the cosine ratio.
 
marinaa11 said:
I just was insecure about my ways of solving this problem I am really thankful for all responds and I gained some new information.
View attachment 24270
It is all about the number 25: regular triangular pyramid's SABC all edges are equal to 8 cm (it is a tetrahedron).
1. Find the surface area of this pyramid.
2. Prove, that pyramid's volume is equal to...
3. Find the sine of the angle that forms the sidewall of this pyramid with the base plane.
4. Find the cosine between the side edge and the base plane.
5. Find the distance from SA and BC.
I am stuck in 25.3 and 25.4
These are mine solutions:
View attachment 24271

These are answers form the book:
View attachment 24272
Click to expand...
Thanks.

For 25.3, you could use the Pythagorean identity [MATH]\sin^2(x) + \cos^2(x) = 1[/MATH] to find the sine. But I don't get their answer.

For 25.4, you could use the same triangle I suggested before as an alternative approach to 25.3, namely SEA, but this time you want angle SAE. I do get their answer here, and your work looks good as far as you showed.
 
Finally, I got 25.4. and I have the correct answer:
136816229_3746595632064184_3060890918468487775_n.jpg

But still, about the sine of the angle that forms the sidewall of this pyramid with the base plane. I tried on the blackhead different solutions to this problem and as Dr. Peterson advised, but I do not get the answer and I still do not understand WHICH angle it is: sidewall would be SAC, the base plane - ABC but how should I "construct" the triangle to find that angle?
 
marinaa11 said:
Finally, I got 25.4. and I have the correct answer:
View attachment 24280

But still, about the sine of the angle that forms the sidewall of this pyramid with the base plane. I tried on the blackhead different solutions to this problem and as Dr. Peterson advised, but I do not get the answer and I still do not understand WHICH angle it is: sidewall would be SAC, the base plane - ABC but how should I "construct" the triangle to find that angle?
Click to expand...
You already constructed it. The angle OES in the triangle OES. It qualifies based on the definition of the angle between planes.
 
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