===========================================================Two men facing a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =120 m, find the distance (in meters) between the two men.
Building's area is negligible
The men are of the same height
They are looking from their feet
Assuming A and B are on the same side of the building.
tan(30) = 120/A
A = 120/tan(30)
tan(60) = 120/B
B = 120/tan(60)
Dif between A and B = 120/tan(30) - 120/tan(60)
= 139m (3sf)
Assuming A and B are on different sides of the building
Dif between A and B = 120/tan(30) + 120/tan(60)
= 277m (3sf)
Assuming A and B are perpendicular to each other
Dif between A and B = Squareroot ((120/tan(30))^2 + (120/tan(60))^2)
= 219m (3sf)
Assuming A and B and building do not lie on the same plane
Infinite answers that are impossible to find.
Sine Rule Explanation
Step 1: Divide triangle ABC into two parts, with a straight line intersecting C and forming a right angle at the intersection with line AB
Step 2: Deduce SinA and SinB, which is opposite/hypotenuse, which is h/b and h/a respectively.
Step 3: Express each equation as a subject of h, forming h= SinAb and h= SinBa
Step 4: SinAb = SinBa , since h = h
Step 5: SinA = (SinBa)/b , bring b over to right side
Step 6: SinA/a = SinB/b , bring a over to left side
Step 7: Hence, SinA/a = SinB/b = SinC/c
Cosine rule explanation
Step 1: CosA = x/b, then make x the subject to be substituted later, which is x = bCosA
Step 2: Use pythagora's theorem on both the left and right hand side right angle triangle.
Left: a^2 = (c-x)^2 + h^2
a^2 = c^2 - 2cx +x^2 + h^2
x^2 + h^2 = 2cx - c^2 + a^2
Right: b^2 = h^2 + x^2
Step 3: Sub right into left
b^2 = 2cx - c^2 + a^2
Step 4: Sub step 1 x into the equation
b^2 = 2bc(cosA) - c^2 + a^2
Step 5: Rearrange the equation
b^2 + c^2 - a^2 = 2bc(CosA)
CosA = (b^2 + c^2 - a^2) / 2bc
By interchanging B and C, we will get similar equations
CosB = (a^2 + c^2 - b^2) / 2ac
CosC = (a^2 + b^2 - c^2) / 2ab
- Analysis of suitability of solution
The answer is 875m which is a suitable as a length of the triangle as it it not too large or small a value. (It is not negative or much larger, e.g. 5 digit figure)
Through the use of a diagram we are able to view the problem in a clearer way and have used Cosine rule to deduce the answer, the values used in the Cosine rule is mostly given in the question, and Cosine rule is a valid way of solving the problem and can be used for any triangle.