Given the following figure, find the lengths 𝐴𝐶 and 𝐵𝐶 and the measure of angle 𝐴𝐵𝐶 in degrees. Give your answers to two decimal places.
In order to answer this question, we need to consider the sine, cosine, and tangent trigonometrical ratios. sin 𝜃 is equal to the opposite divided by the hypotenuse. cos 𝜃 is equal to the adjacent divided by the hypotenuse. And tan 𝜃 is equal to the opposite divided by the adjacent. Labelling the three sides of the right-angled triangle tells us that 𝐵𝐶 is the hypotenuse, the longest side, 𝐴𝐵 is the opposite as it is opposite the 21-degree angle, and 𝐴𝐶 is the adjacent as it is next to or adjacent to the 90-degree and 21-degree angle.
The first part of our question was to work out the length of 𝐴𝐶, labelled 𝑥 on the diagram. 𝐴𝐶 is the adjacent. And 𝐴𝐵 is the opposite. Therefore, we are going to use the tangent ratio. Substituting in these values gives us tan 21 is equal to three divided by 𝑥. Multiplying both sides of the equation by 𝑥 and then dividing both sides by tan 21 gives us 𝑥 is equal to three divided by tan 21. Typing this into the calculator, gives us an answer for 𝑥 of 7.82, to two decimal places. This means that the length 𝐴𝐶 is equal to 7.82.
The second part of our question asked us to work out the length of 𝐵𝐶. As we now know 𝐴𝐶 is 7.82 and 𝐴𝐵 is three, we could use Pythagoras’s theorem to work out the length of 𝐵𝐶. However, in this case, we’re going to continue to use the trigonometrical ratios and use the opposite and the hypotenuse to work out the length 𝐵𝐶. Substituting in our values to the sine ratio gives us sin 21 equals three divide by 𝑦. Once again, using the balancing method allows us to swap the 𝑦 and the sin 21. Therefore, 𝑦 is equal to three divided by sin 21. Typing this into the calculator gives us an answer to two decimal places of 8.37. This means that the length of 𝐵𝐶 in the triangle is 8.37.
The last part of our question asked us to work out the angle 𝐴𝐵𝐶, labelled 𝜃 in the diagram. Now we could use our trigonometrical ratios again to calculate 𝜃. However, we know that angles in a triangle add up to 180 degrees. Therefore, 90 degrees plus 21 degrees plus 𝜃 equals 180 degrees. If we write this out as an equation, we can solve it to find the measure of angle 𝐴𝐵𝐶. 90 plus 21 is 111. Therefore, 111 plus 𝜃 equals 180. Subtracting 111 from both sides of the equation, gives us 𝜃 equal 69 degrees. This means that the angle 𝐴𝐵𝐶 in the triangle is 69 degrees.
This question shows that we can use a mixture of our trigonometrical ratios, Pythagoras’s theorem, and our angle properties to work out all the lengths and angles in a right-angled triangle.