Find sin 𝐶 multiplied by cos 𝐶 given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐴𝐵 equals eight centimeters and 𝐴𝐶 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 𝜃 is equal to the opposite over the hypotenuse. The cos of angle 𝜃 is the adjacent over the hypotenuse. And the tan of angle 𝜃 is equal to the opposite over the adjacent. One way of remembering this is using the acronym SOHCAHTOA.
In this question, the angle 𝜃 is at point 𝐶 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 𝐴𝐶, which is equal to 17 centimeters. This side 𝐴𝐵 is opposite angle 𝐶, and this has a length of eight centimeters. The side 𝐵𝐶 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 𝑥 squared plus 𝑦 squared is equal to 𝑧 squared, where 𝑧 is the length of the hypotenuse or longest side of the triangle. In this question, 𝐵𝐶 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. 𝐵𝐶 squared is therefore equal to 225. And taking the square root of both sides gives us 𝐵𝐶 is equal to 15. As the side length must be positive, we have 𝐵𝐶 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 𝐶 and cos 𝐶. As sin 𝜃 is equal to the opposite over the hypotenuse, sin 𝐶 is equal to eight over 17. In the same way, as the cos of angle 𝜃 is the adjacent over the hypotenuse, the cos of angle 𝐶 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 𝐴𝐵𝐶 as shown, sin 𝐶 multiplied by cos 𝐶 is 120 over 289.