10.3 Solving Right Triangles CHAPTER 10.3 Solving Right Triangles Copyright © 2014 Pearson Education, Inc.

Solving Right Triangles The process of determining the three angles and the lengths of the three sides of a triangle is called solving the triangle. A right triangle can be solved if we know either • the lengths of two sides or • the length of one side and the measure of one acute angle Copyright © 2014 Pearson Education, Inc.

Solving a Right Triangle Given One Side and One Angle Solve the right triangle. If needed, round any answers to one decimal place. Solution m∠B can be found since ∠A and ∠B are complements. m∠B = 90° − m∠A = 90° − 32° = 58° Use either angles A or B in the trigonometric ratios to solve the rest of the triangle. Copyright © 2014 Pearson Education, Inc.

Solving a Right Triangle Given One Side and One Angle is 58 degrees and a is approximately 9.4 units and c is approximately 17.7 units. Copyright © 2014 Pearson Education, Inc.

Solving a Right Triangle Given Two Sides Solve the right triangle. If needed, round any answers to one decimal place. Solution To find f, we can use the Pythagorean Theorem. Copyright © 2014 Pearson Education, Inc.

Solving a Right Triangle Given Two Sides Next, find m∠D or m∠E. We’ll choose m∠D. ∠E and ∠D are complements: m∠E = 90° − m∠D ≈ 90° − 60.9° = 29.1° Copyright © 2014 Pearson Education, Inc.

Angles of Elevation and Depression Many applications of right triangle trigonometry involve the angle made with an imaginary horizontal line. As shown in the figure below, an angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression. Transits and sextants are instruments used to measure such angles. Copyright © 2014 Pearson Education, Inc.

Angles of Elevation and Depression Copyright © 2014 Pearson Education, Inc.

Identifying Angles of Elevation and Depression What is a description of the angle as it relates to the situation shown? a. b. Solution a. ∠1 is the angle of depression from the bird to the person in the hot-air balloon. b. ∠4 is the angle of elevation from the base of the mountain to the person in the hot-air balloon. Copyright © 2014 Pearson Education, Inc.

Problem Solving Using an Angle of Elevation Sighting the top of a building, a surveyor measured the angle of elevation to be 22°. The transit is 5 feet above the ground and 300 feet from the building. Find the building’s height to the nearest whole foot. Copyright © 2014 Pearson Education, Inc.

Problem Solving Using an Angle of Elevation Let a be the height of the part of the building that lies above the transit. The height of the building is the transit’s height, 5 feet, plus a. The trigonometric ratio that will make it possible to find a is tangent. In terms of the 22° angle, we are looking for the side opposite the angle. Also, the side adjacent to the 22° angle is 300 feet, and tangent’s ratio is opp./hyp. Copyright © 2014 Pearson Education, Inc.

Problem Solving Using an Angle of Elevation The height of the part of the building above the transit is approximately 121 feet. The height of the building is h ≈ 5 + 121 = 126 The building’s height is approximately 126 feet. Copyright © 2014 Pearson Education, Inc.

Determining the Angle of Elevation A building that is 21 meters tall casts a shadow 25 meters long. Find the angle of elevation of the sun to the nearest degree. Solution Copyright © 2014 Pearson Education, Inc.

Determining the Angle of Elevation We use a calculator in the degree mode to find m∠A. The display should show approximately 40. The angle of elevation of the sun, m∠A, is approximately 40°. 21 25 Enter Tan-1 ÷ Copyright © 2014 Pearson Education, Inc.