Tony was playing outside with his favorite red ball when an idea came to him. He wanted to find out how far he could throw the ball. Tony threw the ball at a 40 degree angle each time he took steps back. Tony uses trigonometry to determine how far he was from a 15 foot tree that marks his starting point. He figures out that 15/tan(40)=17.88, so he is 17.88 feet away from his starting point. After figuring out how far had gotten, Tony threw his ball once more, but this time, it got stuck in the tree.
Tony's Trigonometric Troubles
Luckily, Tony's friend, Ben, was taking a walk through the neighborhood. Ben heard the commotion in Tony's backyard and decided to check it out.
After Tony told Ben all about what had happened, Ben offered to help him get his ball back. Ben knew that if he could hit the ball with something else, the ball would fall out of the tree. And so, Ben got to work on finding out how far he would need to throw an object so that it would knock down the ball. Based on what Tony told him, Ben used cosine to find that 17.88/cos(40)=23.34, which means that Ben would have throw hard enough for what is thrown to reach a distance of at least 23.34 feet.
Tony and Ben were having so much fun using trigonometry, they even decide to calculate the angle formed by Ben, Tony and the ball. To do this, they used sine to solve for the missing angle. After using the equation θ=sin-1 (17.88/23.34), they found θ=50°.
Because of all the math the boys were doing, they forgot all about the ball that had gotten stuck. From that day forward, Tony and Ben would spend their days solving trigonometric problems, not playing outside with bouncy balls.