the sides of triangle are in arithmetic progession(AP)and of them is 60 find the angles in grade
There are actually infinitely many solutions here. Given that '60' is just any one the angles, that is.
30,60,90, or 20,60,100, so on.
It said that the sides form an AP, not the angles
for angles 30, 60, 90, the sides would be 1 , √3, and 2, which do not form an AP
Oh, my apologies, I considered it to be 'angles are in an AP'
Then again, infinite solutions holds true right?
I suspect you mistyped the question, it should have been "the angles are in arithemtric..." This is an old text problem, in math texts since John Paul Jones was a midshipman.
Let the sides be 60, 60+a, 60+2a
(note that a could be negative, if 60 is not the smallest side)
Then we have
sinA/60 = sinB/(60+a) = sinC/(60+2a)
A+B+C = 180
That is only 3 equations and 4 unknowns, so there are many possible solutions. For example, let the sides be
60,60.1,60.2
Then the sides are very nearly equal, so the angles are also very nearly equal.
Or, let the sides be 60,119,178
Then the triangle is very nearly flat, so two of the angles are tiny, and one is almost 180.
And any solution in between these cases is also possible.
To find the angles of a triangle when the sides are in an arithmetic progression, we can use the cosine rule.
The formula for the cosine rule is given as:
c^2 = a^2 + b^2 - 2ab * cos(C)
where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite side c.
In this case, we are given that one of the sides is 60. Let's assume the common difference of the arithmetic progression is 'd'. So, the lengths of the sides can be represented as:
a = 60 - d
b = 60
c = 60 + d
Now, we can use the cosine rule to find the angle C opposite side c.
c^2 = a^2 + b^2 - 2ab * cos(C)
(60 + d)^2 = (60 - d)^2 + 60^2 - 2(60 - d)(60) * cos(C)
Simplifying the equation will give a quadratic equation, which we can solve to find the value of 'd'. Once we know 'd', we can substitute it back in the side lengths to find the values of a, b, and c.
Once we have all the side lengths, we can use the sine rule or the cosine rule again to find the other angles of the triangle.
Please note that solving the quadratic equation may require further computation, and the answer will depend on the specific values of 'd' you obtain.