### Video Transcript

Find sin πΆ multiplied by cos πΆ given that π΄π΅πΆ is a right triangle at π΅, where π΄π΅ equals eight centimeters and π΄πΆ equals 17 centimeters.

In order to answer this question, we will use our knowledge of trigonometry in right triangles. The trigonometric ratios tell us that the sin of angle π is equal to the opposite over the hypotenuse. The cos of angle π is the adjacent over the hypotenuse. And the tan of angle π is equal to the opposite over the adjacent. One way of remembering this is using the acronym SOHCAHTOA.

In this question, the angle π is at point πΆ of our triangle. We know that the hypotenuse of any right triangle is opposite the right angle and is the longest side. In this case, this is the length π΄πΆ, which is equal to 17 centimeters. This side π΄π΅ is opposite angle πΆ, and this has a length of eight centimeters. The side π΅πΆ is the adjacent side of our triangle.

In order to calculate the length of this side of our triangle, we will use the Pythagorean theorem, which states that π₯ squared plus π¦ squared is equal to π§ squared, where π§ is the length of the hypotenuse or longest side of the triangle. In this question, π΅πΆ squared plus eight squared is equal to 17 squared. Eight squared is equal to 64, and 17 squared is 289. We can then subtract 64 from both sides of this equation. π΅πΆ squared is therefore equal to 225. And taking the square root of both sides gives us π΅πΆ is equal to 15. As the side length must be positive, we have π΅πΆ is equal to 15 centimeters, noting that this triangle is an eight-15-17 Pythagorean triple.

We can now work out the values of sin πΆ and cos πΆ. As sin π is equal to the opposite over the hypotenuse, sin πΆ is equal to eight over 17. In the same way, as the cos of angle π is the adjacent over the hypotenuse, the cos of angle πΆ is 15 over 17. To calculate the product of two fractions, we simply multiply the numerators and then separately multiply the denominators. Eight multiplied by 15 is 120, and 17 multiplied by 17, or 17 squared, is 289. In the right triangle π΄π΅πΆ as shown, sin πΆ multiplied by cos πΆ is 120 over 289.