How to Find the Distance of Y in a Right Triangle

Surveyors can use Pythagorean theorem and trigonometric functions to find the height of a tower.
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All right triangles contain a 90-degree angle. This is the triangle's largest angle, and it is opposite to the longest side. If you have the distances of two sides or the distance of one side plus the measure of one of the right triangle's other angles, you can find the distance of all the sides. Depending on the information available, you can use either the Pythagorean theorem or trigonometric functions to find the length of any side. The study of right triangles finds applications in technical subjects like engineering, architecture and medicine.

    Obtain the proper information to make the calculation. Sketch the right triangle and label the sides --- opposite, adjacent and hypotenuse --- in metric units. Insert the angles in degrees if the question contains that information, or use variable (theta) to label an unknown angle. Write the values for each side; ensure that they are in the same metric units.

    Calculate one side when two sides are given. Calculate the length of a side (Y) using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides. To calculate a length of hypotenuse, calculate adjacent length squared plus opposite length squared, and then calculate the square root of the result with the aid of a calculator.

    To determine the opposite length, calculate hypotenuse length squared minus the adjacent length squared, and then calculate the square root of the result on a calculator. The calculation of adjacent length is similar to the method used to calculate opposite length. The metric unit of your calculated length is the same as those of the given lengths.

    Calculate one side when a side and angle are given. Use the unknown-side label (Y), known-side label and known angle; identify the appropriate trigonometric function relating all three parameters. If the function is cosine, for example, and the unknown label is adjacent, calculate the cosine of the angle with a calculator to obtain a real number. Multiply the real number by the hypotenuse length. The result is the length of the adjacent side, and it has the same unit as the hypotenuse. The use of sine (opposite/hypotenuse) and tangent (opposite/adjacent) functions to find the distance of “Y” is similar to the method used with the cosine function.

    Tips

    • In trigonometry and coordinate geometry, distance and length are synonymous. For simplicity, in labeling right triangles, the side opposite to the 90-degree angle is called hypotenuse, the side containing the 90-degree angle and given angle is called adjacent and the side containing the given angle of interest, but not containing the 90-degree angle, is called opposite.

      Distance of Y refers to an unknown length of a line segment --- adjacent, opposite and hypotenuse --- in a right triangle.

      To convert degrees to radians, multiply the angular measure in degrees by pi, then divide the result by 180 degrees on a calculator.

    Warnings

    • Before using your calculator to determine the values of trigonometric functions, program the calculator to process appropriate angular measures.

      Calculating the values of trigonometric functions using either degrees or radians has the same results, if the calculator is programmed so that the functions process the appropriate arguments.

      Trigonometric functions only take angular values, measured in either degrees or radians.

      Inverse trigonometric functions take real numbers only as their arguments, which normally is the ratio of two sides. The result of an inverse trigonometric function is an angle, and the result of a trigonometric function is a real number.

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