1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Acute Angles and Right Triangles

2. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 2.1 Trigonometric Functions of Acute Angles 2.2 Trigonometric Functions of Non-Acute Angles 2.3 Finding Trigonometric Function Values Using a Calculator 2.4 Solving Right Triangles 2.5Further Applications of Right Triangles 2 Acute Angles and Right Triangles

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Further Applications of Right Triangles 2.5 Bearing ▪ Further Applications

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Bearing There are two methods for expressing bearing. When a single angle is given, such as 164°, it is understood that the bearing is measured in a clockwise direction from due north.

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Example 1 SOLVING A PROBLEM INVOLVING BEARING (METHOD 1) Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to C. Right angles are formed at A and B, so angles CAB and CBA can be found as shown in the figure. Angle C is a right angle because angles CAB and CBA are complementary.

6. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Caution A correctly labeled sketch is crucial when solving bearing applications. Some of the necessary information is often not directly stated in the problem and can be determined only from the sketch.

7. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 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.

8. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Example 2 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2) A ship leaves port and sails on a bearing of N 47º E for 3.5 hr. It then turns and sails on a bearing of S 43º E for 4.0 hr. If the ship’s rate of speed is 22 knots (nautical miles per hour), find the distance that the ship is from port. Draw a sketch as shown in the figure. Choose a point C on a bearing of N 47° E from port at point A. Then choose a point B on a bearing of S 43º E from point C. Because north- south lines are parallel, angle ACD is 47º by alternate interior angles. The measure of angle ACB is 47º + 43º = 90º, making triangle ABC a right triangle.

9. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Example 2 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2) (cont.) Next, use the formula relating distance, rate, and time to find the distances from A to C and from C to B. Now find c, the distance from port at point A to the ship at point B.

10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 3 USING TRIGONOMETRY TO MEASURE A DISTANCE The subtense bar method is a method that surveyors use to determine a small distance d between two points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. Angle θ is measured, then the distance d can be determined. (a)Find d with θ = 1°23′12″ and b = 2.0000 cm. From the figure, we have

11. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Let b = 2. Convert θ to decimal degrees: Example 3 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued)

12. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Since θ is 1″ larger, θ = 1°23′13″ ≈ 1.386944º. (b)How much change would there be in the value of d if θ were measured 1″ larger? Example 3 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued) The difference is

13. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 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 36.7°. He then moves back 50 ft. From the second point, the angle of elevation to the top of the tree is 22.2°. Find the height of the tree to the nearest foot. Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION The figure shows two unknowns: x, the distance from the center of the trunk of the tree to the point where the first observation was made, and h, the height of the tree. Since nothing is given about the length of the hypotenuse, of either triangle ABC or triangle BCD, use a ratio that does not involve the hypotenuse—namely, the tangent.

14. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 In triangle ABC: Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) In triangle BCD: Each expression equals h, so the expressions must be equal.

15. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) Since h = x tan 36.7°, we can substitute. The height of the tree is approximately 45 ft.

16. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Graphing Calculator Solution Superimpose coordinate axes on the figure with D at the origin. The tangent of the angle between the x-axis and the graph of a line with equation y = mx + b is the slope of the line, m. For line DB, m = tan 22.2°. Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) Since b = 0, the equation of line DB is

17. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) The equation of line AB is Since b ≠ 0 here, we use the point A(50, 0) and the point- slope form to find the equation.

18. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Graph y 1 and y 2, then find the point of intersection. The y-coordinate gives the length of BC, or h. Thus, h ≈ 45. Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued)