Question Video: Finding all the Unknowns in a Right Triangle Mathematics

Video Transcript

Given the following figure, find the lengths of 𝐴𝐶 and 𝐵𝐶 and the measure of angle 𝐵𝐴C in degrees. Give your answers to two decimal places.

As the triangle is right angled, we can use the trigonometrical ratios to work out the missing lengths and the missing angles. The three ratios are sine, cosine, and tangent, identified on your calculator by the sin, cos, and tan buttons. 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 side divided by the adjacent. Our first step is to label the triangle. The side opposite the right angle, the longest side, is called the hypotenuse. Side 𝐴𝐵 is the opposite, as it is opposite the 22-degree angle. And side 𝐵𝐶 is adjacent or next to the 22-degree angle.

Let’s first look at how we would work out the length 𝐴𝐶, which we have labelled 𝑥 in this case. As we know the opposite, 𝐴𝐵, and we are trying to work out the hypotenuse, 𝐴𝐶, we’re going to use the sine ratio. Substituting in our values, gives us sin 22 equals four divided by 𝑥. Multiplying both sides of this equation by 𝑥, gives us 𝑥 multiplied by sin 22 equals four. And finally, dividing both sides by sin 22 gives us 𝑥 equals four divided by sin 22.

Ensuring our calculator is in degree mode, typing this into the calculator gives us 𝑥 equals 10.68 to two decimal places. Therefore, the length 𝐴𝐶 in the triangle is equal to 10.68. As we know now two sides of the right-angled triangle, we could use Pythagoras’s theorem to calculate 𝐵𝐶. However, we’re going to stick with the trigonometrical ratios. Length 𝐵𝐶 is the adjacent, length 𝐴𝐵, four, is the opposite. Therefore, we are going to use the tan ratio. Tan 𝜃 equals the opposite divided by the adjacent. If we let the length 𝐵𝐶 equal 𝑦, our equation becomes tan 22 equals four divided by 𝑦.

Rearranging this equation in the same way as before, multiplying by 𝑦, then dividing by tan 22, gives us 𝑦 equals four divided by tan 22. Typing this into the calculator, gives us a value for 𝑦 of 9.90. Therefore, the length of 𝐵𝐶 is equal to 9.90. We now know all three lengths of the triangle and two of the angles, 22 degrees and 90 degrees.

Our final step was to work out the angle 𝐵𝐴𝐶, labeled 𝜃 on the diagram. Now we know that all angles in a triangle add up to 180. Therefore, 90 degrees plus 22 degrees plus 𝜃 must equal 180 degrees. 90 add 22 is 112. So we have 112 plus 𝜃 equals 180. Subtracting 112 from both sides of the equation, gives us a final answer of 𝜃 equals 68 degrees. Therefore, angle 𝐵𝐴𝐶 is equal to 68 degrees.

To summarise, we were asked to find the lengths 𝐴𝐶 and 𝐵𝐶 as well as the angle 𝐵𝐴𝐶. We used the trigonometrical ratios to calculate that 𝐴𝐶 was 10.68. And in a similar way, 𝐵𝐶 was 9.90. We then used the fact that angles in a triangle add up to 180 degrees to work out that 𝐵𝐴𝐶 was equal to 68 degrees.