𝑋𝑌𝑍 is a right-angled triangle at 𝑌, where 𝑋𝑌 is equal to 16.5 centimeters, 𝑌𝑍 is equal to 28 centimeters, and 𝑋𝑍 is equal to 32.5 centimeters. Find the measure of angle 𝑍, giving the answer to the nearest second.
We will begin by sketching the right triangle 𝑋𝑌𝑍. We are told that side 𝑋𝑌 is 16.5 centimeters long. The length of 𝑌𝑍 is 28 centimeters. And the length of 𝑋𝑍 is 32.5 centimeters. We are asked to work out the measure of angle 𝑍. We will do this using our knowledge of right angle trigonometry and the trigonometric ratios, which we can recall using the acronym SOH CAH TOA.
We recall that the longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse. The side opposite angle 𝑍 is known as the opposite. And the side next to angle 𝑍 and the right angle is known as the adjacent. Since we know the lengths of all three sides of our triangle, we can use any one of the three ratios. sin 𝜃 is equal to the opposite over the hypotenuse. cos 𝜃 is equal to the adjacent over the hypotenuse. And tan 𝜃 is equal to the opposite over the adjacent.
In this question, we will use the sine ratio. Substituting in the lengths of the opposite and hypotenuse, we have sin 𝑍 is equal to 16.5 over 32.5. We can then take the inverse sine of both sides of our equation such that 𝑍 is equal to the inverse sin of 16.5 over 32.5. Ensuring that our calculator is in degree mode, we can type in the right-hand side, giving us 30.510237 and so on. This is the answer in degrees.
However, we’re asked to give our answer to the nearest second. We will therefore need to convert this value into degrees, minutes, and seconds. Since there are 60 minutes in one degree, we multiply the decimal part of our answer by 60. This gives us 30.6142 and so on. So our angle is equal to 30 degrees and 30.6142 minutes. There are 60 seconds in one minute. So, once again, we multiply the decimal part of our answer by 60. This gives us 36.854 and so on.
So our angle is equal to 30 degrees, 30 minutes, and 36.854 seconds. Rounding this to the nearest second, we have 30 degrees, 30 minutes, and 37 seconds. This is the measure of angle 𝑍.