Expert: Paul Klarreich - 1/28/2007

Question

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The text above is a follow-up to ...

-----Question-----
I have a triangle   45 degree and the bottom is 173 ft, Another angle is 60.  How can I figure out how long the other two sides are?
-----Answer-----
Questioner:   Layne
Category:  Advanced Math
Private:  no
 
Subject:  Finding Length of two sides of triangle
Question:  I have a triangle   45 degree and the bottom is 173 ft, Another angle is

60.  How can I figure out how long the other two sides are?

Hi, Layne,

Use the Law of Sines, which says that in triangle ABC, using the convention that side a is opposite angle A, etc.:

 a       b       c
----- = ----- = -----
sin A   sin B   sin C

Also, use the sum-of-angles property to determine that the third angle is 75 degrees.

NOW determine which of the angles is opposite your 173-ft side. You didn't say which it is, so we can't go any further in the problem.  

Suppose it is opposite angle A.  Then your proportions (yes, the above law actually has three equations built into it) say:

173      b       c
----- = ----- = -----
sin A   sin B   sin C

where A,B,C are the 45, 60, and 75, in whatever is the correct layout.

Thank you, I apologize that I forgot to that the 173 ft line is on point a and b...a being the 45 degree angle...b being at the 60 degree angle.

I have had the response that give me and answer "hit your calculator to find the tangent of the 60degree angle"  however, How do we figure this out without using a calculator?

Answer
Questioner:   Layne
Category:  Advanced Math
Private:  no
 
Subject:  Finding Length of two sides of triangle
Question:  I have a triangle   45 degree and the bottom is 173 ft, Another angle is 60.  How can I figure out how long the other two sides are?
================== OLD STUFF ================
Hi, Layne,

Use the Law of Sines, which says that in triangle ABC, using the convention that side a is opposite angle A, etc.:
 a       b       c
----- = ----- = -----
sin A   sin B   sin C

Also, use the sum-of-angles property to determine that the third angle is 75 degrees.

NOW determine which of the angles is opposite your 173-ft side. You didn't say which it is, so we can't go any further in the problem.  

Suppose it is opposite angle A.  Then your proportions (yes, the above law actually has three equations built into it) say:

173      b       c
----- = ----- = -----
sin A   sin B   sin C

where A,B,C are the 45, 60, and 75, in whatever is the correct layout.
================ END OF OLD STUFF ==================
............................
Thank you, I apologize that I forgot to that the 173 ft line is on point a and b...a being the 45 degree angle...b being at the 60 degree angle.

>> You MUST be very careful about the notation.  There is a difference between 'a' and 'A' -- the convention is to use small letters for the sides and capitals for the angles.  Then side BC is the same as side a, but A means angle A.

I have had the response that give me and answer "hit your calculator to find the tangent of the 60 degree angle"  however, How do we figure this out without using a calculator?

>> What do tangents have to do with this example?

However, trigonometric functions of 'special' angles (30, 45, and 60) and  can be computed in 'exact' form without using a calculator:

For 30 or 60 degrees, draw a right triangle (not the triangle in your example) with angles of 30, 60, and 90.  Now assign a length of 2 to the hypotenuse and you find that:

the leg opposite 30 is 1/2 the hypotenuse, or 1.
the leg opposite 60 is 1/2 the hypotenuse times sqrt(3), or sqrt(3).

Then:  sin 30 = cos 60 = 1/2, and
cos 30 = sin 60 = sqrt(3)/2.

For 45 degrees, draw a right triangle with angles of 45, 45, and 90.  Now assign a length of 2 to the hypotenuse and you find that each leg is equal to sqrt(2).  

Then: sin 45 = cos 45 = sqrt(2)/2.

Those are exact values, even though they are not in decimal format.  To get them in decimal format, you will still need the calculator.

In your example, if  AB = 173 (don't write ab, remember), then that is 'c' in the Law of Sines, and if A = 45,  B = 60, then C = 75:

 a        b       173
------ = ------ = ------
sin 45   sin 60   sin 75

To find a (remember, that's not angle A), write this piece of the Law:

 a        173
------ =  ------
sin 45    sin 75
   173 sin 45
a = ----------
     sin 75

You will need the calculator for sin 75.  There IS a way to find sin 75 in exact form, but I don't think you are ready for it.

To find b:

 b       173
------ = ------
sin 60   sin 75
 
and do the same.