How to Determine the Height of a Tree

Which trig function should you use to determine the height of a tree? Suppose you're flying a kite, and it gets caught at the top of a tree. You've let out all 100 feet of string for the kite, and the angle that the string makes with the ground (the angle of elevation) is 75 degrees.

Instead of worrying about how to get your kite back, you wonder, “How tall is that tree?”

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To find a solution to your quandary, follow these steps:

  1. Identify the parts of the right triangle that you can use to solve the problem.

    The hypotenuse of the right triangle is the length of the string. The side opposite the 75-degree angle is what you're solving for; call it x.

  2. Determine which trig function to use.

    The hypotenuse and opposite side are part of the sine ratio.

  3. Write an equation with the trig function; then insert the values that you know.

    The 75-degree angle isn't one of the more-common angles, so use a scientific calculator or one of the tables in the Appendix to obtain a value for the sine, correct to three decimal places. The sine of 75 degrees is about 0.966, the hypotenuse is 100 feet, and the opposite side is what is unknown.

    image1.jpg
  4. Solve the equation.

    Cross-multiplying, you get

    image2.jpg

    The tree is over 96 feet tall. Lots of luck retrieving the kite.

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