How To Find The Height Of A Triangle Using Trigonometry

ESTIMATION OF TREE HEIGHT: Right TRIANGLE TRIGONOMETRY

Introduction: Scientists studying a forest ecosystem over a long period of time may record measurements of trees for a number of variables, including each tree'south bore at chest elevation, tiptop of the lowest living branch, awning cover, etc. One aspect of a tree's growth that can be hard to measure is tree acme. Woods researchers sometimes use a piece of equipment consisting of telescoping components, which are extended until the tip reaches the same superlative every bit the tree top (this requires a second researcher standing at a distance from the tree to determine when the tip is at the right meridian). This method can be cumbersome, as the equipment is bulky and the measurements require two people.

Importance: The measurement of tree growth in a forest over fourth dimension provides of import information nigh the dynamics of that ecosystem. Growth rates tin reverberate, among other things, differing availability of water, carbohydrates, or nutrients in unlike sites or from season to season or year to year.

Question: Is in that location an efficient style to measure out tree top, without heavy equipment and multiple people?

Variables:

q	an acute angle of a right triangle
x	side of a triangle adjacent to the angle q
y	side of a triangle reverse the bending q
z	the hypotenuse of a triangle

Methods: Trigonometry is a branch of mathematics dealing with measurements of the angles and sides of triangles, and functions based on these measurements. The three basic trigonometric functions that we are concerned with here (sine, cosine, and tangent) are ratios of the lengths of two sides of a triangle. These ratios are the trigonometric functions of an angle, theta, such that

where q (theta) is the angle of interest, "opp" is the length of the side of the triangle opposite q (y), "hyp" is the length of the hypotenuse (z), and "adj" is the length of the side side by side to the angle q (x), as illustrated on the triangle below.

If we know the lengths of two sides of a right triangle (call back from geometry that a correct triangle has one bending that is 90 degrees [a correct angle]), we can calculate the length of the third side using the Pythagorean theorem (opp² + adj² = hyp², or y ² + 10 ² = z ²). For the triangle below, the side opposite q is three units in length, and the side adjacent to q is 1.5 units in length.

The length of the hypotenuse is calculated three² + 1.5ⁱⁱ = hyp² =eleven.25. Taking the square root of 11.25, we find that the hypotenuse is approximately 3.35 units long. We now know the lengths of each of the sides of the triangle, and tin can use these to find values for the trigonometric functions:

Simply what can we do with this information? Well, for one matter we can use information technology to find q by taking the inverse of whatsoever of the functions (for our purposes hither can do this using a calculator). Doing this we encounter that q is an angle of approximately 63.4 degrees.

The drawing below shows a forester measuring a tree'south height using trigonometry. Assuming that the tree is at a correct bending to the plane on which the forester is standing, the base of operations of the tree, the top of the tree, and the forester form the vertices (or corners) of a right triangle. The forester measures his or her distance from the base of operations of the tree, and then uses a clinometer (a pocket-sized instrument that measures inclination, or angle of elevation) to look at the top of the tree and determine q .

Interpretation: In this state of affairs, rather than knowing the lengths of all of the sides, we know q and the length of the adjacent side (x), and are interested in determining the length of the opposite side (y, the height of the tree). Which of the three trigonometric functions deals with the adjacent and opposite sides?

We know q , and nosotros know "adj" (or x); multiplying both sides of the equation by "adj" or 10 and substituting our known values, we get

The tree is approximately 44 feet alpine.

Conclusions: Trigonometry has many real-world applications. In the example here, information technology tin be used then that forest researchers don't have to carry around boosted equipment and are able to collect the necessary data for calculating tree heights speedily and efficiently in the field.

Additional Questions:
one) You know from your geometry class that all the angles of a triangle must sum to 180 degrees. Therefore, the 3rd angle from the tree and forester triangle must be (180 degrees - 90 degrees - 31.eight degrees) = 58.2 degrees. Verify this using the 3 trigonometric functions (annotation that for this bending, y is the adjacent side and x is the opposite side).

2) What would the bending of pinnacle ( q ) be if the tree was 100 ft. tall?

Sources:
Larson, R. E., R. P. Hostetler, and B. H. Edwards. 1993. Precalculus: A Graphing Approach. D. C. Heath and Company, Lexington, MA.

Waring, R. H. and Due west. H. Schlesinger. 1985. Forest Ecosystems: Concepts and Direction. Academic Printing, Inc. San Diego, CA.