Find sin 𝐵, given 𝐴𝐵𝐶 is a right-angled triangle at 𝐶, where 𝐴𝐵 equals 17 centimeters and 𝐵𝐶 equals 15 centimeters.
So we have a right-angled triangle, in which we’ve been given the lengths of two of its sides. We’re asked to find the value of sin 𝐵, which is a trigonometric ratio in reference to the angle 𝐵. Let’s recall the definition of sine.
In a right-angled triangle, the sine of an angle, in this case angle 𝐵, is equal to the ratio of the opposite side divided by the hypotenuse. Let’s look at which side is which for the triangle in this question. Side 𝐴𝐵 is the hypotenuse because it’s opposite the right angle. With respect to angle 𝐵, side 𝐵𝐶 is the adjacent because it’s between angle 𝐵 and the right angle.
Again, with respect to angle 𝐵, side 𝐴𝐶 is the opposite. So this means that in this triangle, the ratio of sin 𝐵 is going to be found by dividing the length of 𝐴𝐶 by the length of 𝐴𝐵. Now, we know the length of 𝐴𝐵; it’s given in the diagram 17 centimeters, but we don’t know the length of 𝐴𝐶. So in order to calculate this sine ratio, we need a method of finding the length of 𝐴𝐶.
Let’s think about what else we know about right-angled triangles. In this triangle, we know the length of two of the sides and we want to calculate the length of third, which means this is the perfect setting to apply the Pythagorean theorem. Remember the Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
In this triangle, this means that 𝐴𝐶 squared plus 15 squared is equal to 17 squared. Now, this is an equation that we can solve in order to find the length of 𝐴𝐶. Evaluating 15 squared and 17 squared gives 𝐴𝐶 squared plus 225 is equal to 289. Next, we subtract 225 from both sides and this gives 𝐴𝐶 squared is equal to 64. Finally, we square root both sides of this equation 𝐴𝐶 is equal to the square root of 64, which is eight.
So now, we know the length of the third side of the triangle. You may also have spotted this quite quickly, if you’re familiar with your Pythagorean triples because eight, 15, 17 is an example of one. In any case, we now know the length of 𝐴𝐶.
So let’s return to the focus of this question, which was to calculate the trigonometric ratio sin of 𝐵. We found that it was equal to 𝐴𝐶 over 17. We now know that 𝐴𝐶 is equal to eight. Therefore, sin of angle 𝐵 is equal to eight over 17.