Question Video: Finding the Measure of an Angle in a Right-Angled Triangle given the Lengths of Its Sides | Nagwa Question Video: Finding the Measure of an Angle in a Right-Angled Triangle given the Lengths of Its Sides | Nagwa

Question Video: Finding the Measure of an Angle in a Right-Angled Triangle given the Lengths of Its Sides Mathematics • Third Year of Preparatory School

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𝑋𝑌𝑍 is a right-angled triangle at 𝑌, where 𝑋𝑌 = 16.5 cm, 𝑌𝑍 = 28 cm, and 𝑋𝑍 = 32.5 cm. Find the measure of ∠𝑍 giving the answer to the nearest second.

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Video Transcript

𝑋𝑌𝑍 is a right-angled triangle at 𝑌, where 𝑋𝑌 equals 16.5 centimeters, 𝑌𝑍 equals 28 centimeters, and 𝑋𝑍 equals 32.5 centimeters. Find the measure of angle 𝑍 giving the answer to the nearest second.

We will begin by sketching the right triangle from the information given. We know that the right angle is at point 𝑌. And the lengths of the three sides are 16.5, 28, and 32.5 centimeters. We are asked to calculate the measure of the angle at 𝑍.

We can do this using our knowledge of the trigonometric ratios in right triangles. We know that 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. One way of recalling these ratios is using the acronym SOHCAHTOA.

We know that the longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse. The side that is opposite the angle we are working with, in this case, 𝑋𝑌, is known as the opposite. The third side, which is next to the angle we are dealing with and the right angle is known as the adjacent. Since we know the lengths of all three sides of the triangle, we can use any one of the three ratios. In this question, we will use the cosine ratio.

Substituting in our values, we have that cos of angle 𝑍 is equal to 28 over 32.5. Taking the inverse cos of both sides of this equation, we have 𝑍 is equal to the inverse cos of 28 over 32.5. Ensuring that our calculator is in degree mode, we can type this in giving us an answer of 𝑍 is equal to 30.510237 and so on.

We are asked to give our answer to the nearest second. We can do this directly on the calculator by pressing the button shown. This gives us an answer of 30 degrees, 30 minutes, and 36.85 seconds, which rounding to the nearest second is 37 seconds.

An alternative method here would be to recall that there are 60 minutes in one degree. We can therefore multiply the part of our answer after the decimal point by 60, giving us 30.6142 and so on minutes. As there are 60 seconds in a minute, we can then multiply the part of this answer after the decimal point by 60, giving us 37 seconds to the nearest second. The measure of angle 𝑍 is 30 degrees, 30 minutes, and 37 seconds to the nearest second.

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