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Question Video: Finding the Sine and Cosine of an Angle in a Right Triangle given the Lengths of Two Sides Mathematics • 10th Grade

Find the value of 2 sin 𝑋 cos 𝑋 given that π‘‹π‘Œπ‘ is a right triangle at π‘Œ, where π‘‹π‘Œ = 10 cm and 𝑋𝑍 = 26 cm.

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Video Transcript

Find the value of two sin 𝑋 cos 𝑋 given that π‘‹π‘Œπ‘ is a right triangle at π‘Œ, where π‘‹π‘Œ equals 10 centimeters and 𝑋𝑍 equals 26 centimeters.

In order to answer this question, we will begin by sketching the triangle π‘‹π‘Œπ‘. We are told that the right angle is at point π‘Œ, that π‘‹π‘Œ is equal to 10 centimeters, and 𝑋𝑍 is equal to 26 centimeters. We are asked to find the value of two sin 𝑋 cos 𝑋. And we will do this using our knowledge of the trigonometric ratios.

We know that the sin of angle πœƒ is equal to the opposite over the hypotenuse. The cos of angle πœƒ is equal to the adjacent over the hypotenuse. And the tan of angle πœƒ is equal to the opposite over the adjacent. These three ratios are often recalled using the acronym SOHCAHTOA.

In this question, the angle we are dealing with is at the point 𝑋. This means that side length π‘Œπ‘ will be the opposite, as it is opposite angle 𝑋. The side length π‘‹π‘Œ is the adjacent, as this is adjacent or next to angle 𝑋 and the right angle. The side length 𝑋𝑍 is the hypotenuse, which is opposite the right angle and is the longest side of our triangle.

As we know the length of the adjacent and hypotenuse, we can work out the value of cos 𝑋. This is equal to 10 over 26. Dividing the numerator and denominator by two, this simplifies to five over 13 or five-thirteenths. In order to calculate the sin of angle 𝑋, we firstly need to work out the length of the opposite side. We can do this using the Pythagorean theorem, which states that π‘Ž squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse of our triangle.

In this question, we have π‘Œπ‘ squared plus 10 squared is equal to 26 squared. As 10 squared is equal to 100 and 26 squared is 676, this simplifies to π‘Œπ‘ squared plus 100 is 676. We can then subtract 100 from both sides of this equation. Finally, square rooting both sides and taking the positive value as we are measuring a length, we have π‘Œπ‘ is equal to 24 centimeters.

The side length opposite angle 𝑋 is equal to 24 centimeters. The sin of angle 𝑋 is therefore equal to 24 over 26. Once again, this can be simplified to 12 over 13, or twelve thirteenths. We can now calculate two multiplied by sin 𝑋 multiplied by cos 𝑋. This is equal to two multiplied by twelve thirteenths multiplied by five-thirteenths. Multiplying the numerators here gives us 120, and multiplying the denominators gives us 169.

The value of two sin 𝑋 cos 𝑋 is 120 over 169.

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