A kite, which is at a perpendicular height of 44 meters, is attached to a string inclined at 60 degrees to the horizontal. Find the length of the string accurate to one decimal place.
We have a kite with a perpendicular height. That means the height forms a right angle with the horizontal. And the height is 44 metres. The kite string also makes an angle with the horizontal. That angle is 60 degrees. The length of the string is unknown. We’ll call that value 𝑙. What we see is that the kite string, the horizontal and the perpendicular height, together form a right angle. And if we use 60 degrees as our angle, we know an opposite side length. And we’re interested in the hypotenuse side length.
We know that the length of the string 𝑙 is the hypotenuse because it’s the side length opposite the right angle. In right-angle trigonometry, the ratio opposite over hypotenuse is the sine relationship. Sin of 𝜃 equals the opposite over the hypotenuse. Sin of 60 degrees equals 44 over 𝑙.
To solve for 𝑙, we need to get out of the denominator. We can do that by multiplying both sides of the equation by 𝑙 over one. On the left, we have 𝑙 times sin of 60 degrees. And on the right, the 𝑙s cancel out, leaving us with 44 over one or just 44. To get 𝑙 by itself, we’ll divide both sides of the equation by sin of 60 degrees. On the left side, the sins cancel out, leaving us with 𝑙 is equal to 44 divided by the sin of 60 degrees.
The sin of 60 degrees is a very common sin. So you might remember that it’s the square root of three over two. We can substitute the square root of three over two and say 44 divided by the square root of three over two. Dividing by a fraction is the same thing as multiplying by its reciprocal. 44 times two equals 88, divided by the square root of three. The length of the string is 88 divided by the square root of three.
Since we need this accurate to one decimal place, we’re going to have to put it into the calculator. In your calculator, you could’ve entered 44 divided by sin of 60 degrees. Or since we’ve solved it a little bit further, we can now enter 88 divided by the square root of three. Either way, the calculator will return 50.806 continuing. If we want to round to the first decimal place, there’s an eight in that position. And we look to the right of the first decimal place. We look to the second decimal place, to the deciding digit. It’s a zero. It’s less than five. And therefore, we round down.
We say that the length of the string is 50.8. We’re measuring in metres, so 50.8 metres.