Chapter 8: Trigonometric Functions and Applications

2. Chapter 8: Trigonometric Functions and Applications 8.1 Angles, Arcs, and Their Measures 8.2 The Unit Circle and Its Functions 8.3 Graphs of the Sine and Cosine Functions 8.4Graphs of the Other Circular Functions 8.5 Functions of Angles and Fundamental Identities 8.6 Evaluating Trigonometric Functions 8.7Applications of Right Triangles 8.8 Harmonic Motion

3. 8.7 Applications of Right Triangles • Significant Digits • Represents the actual measurement. • Most values of trigonometric functions and virtually all measurements are approximations.

4. 8.7 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. 8.7 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,

6. 8.7 Angles of Elevation or Depression The angle of elevation from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X. The angle of depression from point X to point Y (below X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X.

7. 8.7 Angles of Elevation or Depression Solving an Applied Trigonometry Problem 1. Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. 2. Use the sketch to write an equation relating the given quantities to the variable. 3. Solve the equation, and check that your answer makes sense.

8. 8.7 Solving a Problem Involving Angle of Elevation Example The length of the shadow of a building 34.09 meters tall is 3.62 meters. Find the elevation of the sun. The angle of elevation of the sun is 42.18o.

9. 8.7 Bearing There are two methods for expressing bearing. When a single angle is given, it is understood that the bearing is measure in a clockwise direction from the north.

10. 8.7 Solving a Problem Involving Bearing (First Method) Example Radar stations A and B are on an east-west line, with A west of B, 3.70 km apart. Station A detects a plane at C, on a bearing of 61.0o. Station B simultaneously detects the same plane, on a bearing of 331.0o. Find the distance from A to C.

11. 8.7 Solving a Problem Involving Bearing (First Method) Solution Draw a sketch. Since a line drawn due north is perpendicular to an east-west line, right angles are formed at A and B, so angles CAB and CBA can be found. Angle C is a right angle. Find distance b by using the cosine function.

12. 8.7 Bearing The second method for expressing bearing starts with a north-south line and uses an acute angle to show the direction, either east or west, from this line. Either N or S always come first, followed by an acute angle, and then E or W.

13. 8.7 Solving a Problem Involving Bearing (Second Method) Example The bearing from A to C is S 52o E. The bearing from A to B is N 84o E. The bearing from B to C is S 38o W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the distance from A to C.

14. 8.7 Solving a Problem Involving Bearing (Second Method) Solution Make a sketch. First draw the two bearings from point A. Then choose a point B on the bearing N 84o E from A, and draw the bearing to C. Point C will be located where the bearing lines from A and B intersect. The distance from A to B is 250(2.4) = 600 miles.

15. 8.7 Solving a Problem Involving Bearing (Second Method) Solution (continued) To find b, the distance from A to C, use the sine function.

16. 8.7 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   0.0000811º.

17. 8.7 Calculating the Distance to a Star Example One of the nearest stars is Alpha Centauri, which has a parallax of   0.000212º. • 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.

18. 8.7 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

19. 8.7 Solving a Problem Involving Angle of Elevation 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.

20. 8.7 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.

21. 8.7 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).

22. 8.7 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).