Two men are standing in front of a minaret 𝐴𝐷 at the points 𝐵 and 𝐶, respectively, where the distance between them is 25.4 metres. Find the height of the minaret giving the answer to one decimal place.
The first thing we can do is calculate the size of the measure of the obtuse angle at 𝐵. We know that angles on a straight line add to 180 degrees. So we can subtract 64 from 180. That tells us that the measure of the obtuse angle at 𝐵 is 116 degrees.
Now that we know the size of two of the angles in triangle 𝐴𝐵𝐶, we can calculate the size of the third angle. Remember this little symbol means minutes. It’s base 60. This means that 34 degrees and 48 minutes is the same as 34 and forty-eight sixtieths of a degree. Forty-eight sixtieths is the same as 0.8. So 34 degrees and 48 minutes is the same as 34.8 degrees.
Angles in a triangle sum to 180 degrees. So we can calculate the size of angle 𝐶𝐴𝐵 by subtracting 34.8 and 116 from 180. The measure of the angle at 𝐴 — that’s angle 𝐶𝐴𝐵 — is 29.2 degrees.
So we have a non-right-angled triangle, for which we know the size of three of its angles and the length of one of its sides. We can use the law of sines then to find the size of the shared length between the two triangles — that’s the side 𝐴𝐵.
Remember the side opposite the angle 𝐴 is lowercase 𝑎, the side opposite the angle 𝐵 is lowercase 𝑏, and the side opposite angle 𝐶 is lowercase 𝑐. Since we’re trying to calculate the length of the side 𝐴𝐵, that’s lowercase 𝑐, we’ll use the first of the formulae for the law of sines.
In fact, it doesn’t really matter which of these we choose to use. However, since we’re trying to find the length of the side rather than an angle by using the first formula, we’re reducing the amount of rearranging required. The second formula is easier to use when we’re trying to calculate a missing angle.
We know the length of the side 𝑎 and we’re trying to calculate the side 𝑐. We’re not interested in 𝑏 at all. So we’re going to use the formula 𝑎 over sin 𝐴 is equal to 𝑐 over sin 𝐶. We can substitute the values from our triangle into this formula. That gives us 25.4 over sin of 29.2 is equal to 𝑐 over sin of 34.8.
Then, to solve this equation, we’ll multiply both sides by sin of 34.8. That gives us 25.4 over sin of 29.2 multiplied by sin of 34.8. Popping that into our calculator, we get that 𝑐 is 29.7137 and so on metres.
We won’t round this number just yet. Instead, we’ll use the exact number in our next calculation. This will reduce any possible errors from rounding too early. We now know the length of 𝐴𝐵: it’s 29.7137 metres. We’re trying to find the length of the side 𝐴𝐷.
Triangle 𝐴𝐵𝐷 is a right-angled triangle with a hypotenuse of length 29.7137. Its opposite side is the side labelled 𝑥. It’s the side opposite the angle of 64 degrees. Since we know the length of the hypotenuse and we’re trying to calculate the length of the opposite side, we can use the sine ratio, where sin of 𝜃 is equal to opposite divided by hypotenuse.
Substituting the values from our triangle into the formula gives us sin of 64 equals 𝑥 over 29.7137. We can solve this equation by multiplying both sides by 29.7137. 𝑥 is, therefore, equal to sin of 64 multiplied by 29.7137. That’s 29.706.
Correct to one decimal place, the height of the minaret is 29.7 metres.