The height of the steeple can be calculated by using trigonometry. Trigonometry is an ancient branch of mathematics which studies relationships between the sides and angles of triangles in two-dimensional space. In this case, we have a right angled triangle with Jack and Jill being on either side and the steeple at the vertex (point) where both their lines of sight intersect at 90 degrees. To calculate the height of this steeple, we must use basic trigonometric functions such as sine, cosine, tangent and their inverse counterparts to solve for unknown side lengths.

### Jack and Jill are on Either side of a church and 50 m apart. Jack sees the top of the steeple at 40° and Jill sees the top of the steeple at 30°. How high is the steeple?

To calculate the height of the church’s steeple using Jack and Jill’s angle measurements at 40° and 30° respectively, firstly one must draw a diagram of what they are seeing from each person’s perspective. This will include identifying that these angles are complementary since they add together to make up 90 degrees; meaning that if one subtracts Jack’s angle measurement from 90° then you will get Jill’s angle measurement which is 50° – 40°= 10°.

Once we have identified all our angle measurements within our triangle, we can now use some basic trigonometry formulas to calculate our unknown side lengths: Sine Rule = Opposite/Sin(Angle), Cosine Rule = Hypotenuse^2 = Opposite^2 + Adjacent^2 or Tangent = Opposite/Adjacent. With these formulas it is possible to find out how long each edge connecting two vertices is for any given triangle based off knowing just two sides or an angle measurement along with another side length – this requires algebraic manipulation however it allows us to reduce our number of unknown variables until there are no more left over!

In terms of solving this particular triangular question, once all three angles have been assigned (40º , 50º , 30º) then it becomes possible to identify either two sides or even both pieces required for calculating hypotenuse squared under cosine rule equation given above; thus in turn providing us with exact answer for our desired elevation (height) value which would be 44 meters tall! The calculation works like this: First determine adjacent side lengths by applying sine rule formula on Angle A & B (40 & 50 degree): sin(40)/a=sin(50)/b ; Solve for b : b=(sin(50)*a)/sin(40). Then substitute values into Cosine Rule equation: c^2=a^2+b^2 ; Solve c : c=sqrt[a^2+(b*sin(50))/sin(40)]. Finally plug c into Pythagorean Theorem equation since right triangle always satisfies pythagoras theorem : d=(c^2-a^2) ; Solution d=44 m tall!

Therefore it has been established that when observing a church’s steeple from different perspectives yet still being able to come up with same elevation value thanks to trigonometric principles like sines rules & cosines rules combined alongside Pythagorean Theorem equations – one can accurately conclude its estimated height as 44m tall overall!