Applications of Right Triangles Significant Digits Represents the actual measurement. Most values of trigonometric functions and virtually all measurements are approximations.

5.7 Solving a Right Triangle Given an Angle and a Side Example Solve the right triangle ABC, with A = 34º 30 and c = 12.7 inches. Solution Angle B = 90º – A = 89º 60 – 34º 30 = 55º 30. Use given information to find b.

5.7 Solving a Right Triangle Given Two Sides Example Solve right triangle ABC if a = 29.43 centimeters and c = 53.58 centimeters. Solution Draw a sketch showing the given information. Using the inverse sine function on a calculator, we find A  33.32º. B = 90º – 33.32º  56.68º Using the Pythagorean theorem,

5.7 Calculating the Distance to a Star In 1838, Friedrich Bessel determined the distance to a star called 61 Cygni using a parallax method that relied on the measurement of very small angles. You observe parallax when you ride in a car and see a nearby object apparently move backward with respect to more distance objects. As the Earth revolved around the sun, the observed parallax of 61 Cygni is   .0000811º.

5.7 Calculating the Distance to a Star Example One of the nearest stars is Alpha Centauri, which has a parallax of   .000212º. Calculate the distance to Alpha Centauri if the Earth-Sun distance is 93,000,000 miles. A light-year is defined to be the distance that light travels in 1 year and equals about 5.9 trillion miles. Find the distance to Alpha Centauri in light-years.

5.7 Calculating the Distance to a Star Solution Let d be the distance between Earth and Alpha Centauri. From the figure on slide 8-46, (b) This distance equals

5.7 Solving a Problem Involving Angle of Elevation Angles of Elevation or Depression Example Francisco needs to know the height of a tree. From a given point on the ground, he finds that the angle of elevation to the top of the tree is 36.7º. He then moves back 50 feet. From the second point, the angle of elevation is 22.2º. Find the height of the tree.

5.7 Solving a Problem Involving Angle of Elevation Analytic Solution There are two unknowns, the distance x and h, the height of the tree. In triangle ABC, In triangle BCD, Each expression equals h, so the expressions must be equal.

5.7 Solving a Problem Involving Angle of Elevation We saw above that h = x tan 36.7º. Substituting for x, Graphing Calculator Solution Superimpose the figure on the coordinate axes with D at the origin. Line DB has m = tan 22.2º with y-intercept 0. So the equation of line DB is y = tan 22.2º x. Similarly for line AB, using the point-slope form of a line, we get the equation y = [tan 36.7º](x – 50).

5.7 Solving a Problem Involving Angle of Elevation Plot the lines DB and AB on the graphing calculator and find the point of intersection. Rounding the information at the bottom of the screen, we see that h  45 feet. Line DB: y = tan 22.2º x Line AB: y = [tan 36.7º](x – 50).