A bridge is to be constructed over a canyon stretching from point 𝐴 to point 𝐵 as seen in the given figure. A surveyor stands at point 𝐶, 30 yards from point 𝐴, at the edge of the canyon. They measured that the measure of the angle 𝐶𝐴𝐵 is 70 degrees and the measure of the angle 𝐴𝐵𝐶 is 28 degrees. Work out the length of the bridge.
There’s rather a lot of extra stuff in this question. And it’s really useful to minimalize what we’ve been given by sketching a less detailed diagram, as you can see above. Our next step is to make a decision as to which rule we need here.
Firstly, we can see it as a non-right-angled triangle, which instantly removes basic trigonometry, in other words SOHCAHTOA and Pythagoras’s theorem. That narrows it down to the sine rule or cosine rule, since we know we definitely aren’t interested in the area of the triangle we sketched.
The number of angles is what will help us make the final decision here. The cosine rule has only one angle in the formula. Therefore, we use the cosine rule when we are either given only one angle and need to find a side or when we know no angles but do need to find one of them. The sine rule it is then. Remember, we actually only ever need to find two parts of this. And we can choose the orientation depending on what we’re trying to find. More on that in a moment.
We know the length of the side we’ve labeled as lowercase 𝑏 to be 30 yards and its corresponding angle pair at 𝑏 to be 28 degrees. We’re trying to find the length of the bridge which we have labeled as lowercase 𝑐 in our diagram. This means we need the size of its angle pair at 𝑐.
To calculate this, we use the fact that angles in a triangle add to 180 degrees. 180 minus 70 and 28 is 82 degrees. Now let’s substitute these values into our formula. We’re going to choose the formula on the right-hand side because our unknown is a side, as opposed to an angle. This will make our rearranging a bit less awkward. 30 over sin 28 is equal to 𝑐 over sin 82.
To rearrange, we multiply both sides by sin 82, giving us 𝑐 equals 30 over sin 28 times sin 82. 𝑐 is therefore equal to 63.28 yards, correct to two decimal places.