Group 1 (Shaun, Desiree, Owen, Ryan)

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

Question 1 

  1. 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.


  1. A visual representation would be much appreciation. Please include a diagram.

  2. You did not really specify assumptions. What are your assumption.

    You did not specify who/what A and B are. Do they represent the 2 men?

    You can add some pictures to illustrate what you are talking about. Especially for the last assumption where you stated that there are infinite answers. Why are there infinite answers?

    Working is very concise and easy to understand.

    - Group 4 (Mason, Kai En, Crystal, Wai Kit, Farrell)