Trigonometry and Angles of Elevation and Depression CHAPTER 8.4 AND 8.5

Trigonometry Trigonometry is the study of triangle measurement A trigonometric ratio is a ratio of the lengths of two sides of a right triangles.

Example 1 Find Sine, Cosine, and Tangent Ratios A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 Find Sine, Cosine, and Tangent Ratios B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 Find Sine, Cosine, and Tangent Ratios C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 Find Sine, Cosine, and Tangent Ratios D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 Find Sine, Cosine, and Tangent Ratios E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 Find Sine, Cosine, and Tangent Ratios F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer:

Example 1 A. Find sin A. A. B. C. D.

Example 1 B. Find cos A. A. B. C. D.

Example 1 C. Find tan A. A. B. C. D.

Example 1 D. Find sin B. A. B. C. D.

Example 1 E. Find cos B. A. B. C. D.

Example 1 F. Find tan B. A. B. C. D.

Example 2 Use Special Right Triangles to Find Trigonometric Ratios Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse. The side adjacent to the 60° angle has a measure of x.

Example 2 Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D.

Example 3 Estimate Measures Using Trigonometry EXERCISING A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

Example 3 A.1 in. B.11 in. C.16 in. D.15 in. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?

Try it! Find each length using trigonometry. Pg 570 #3A and 3B

Example 4 Find Angle Measures Using Inverse Trigonometric Ratios Use a calculator to find the measure of  P to the nearest tenth.

Example 4 A.44.1° B.48.3° C.55.4° D.57.2° Use a calculator to find the measure of  D to the nearest tenth.

Try it! Find each angle measure. Pg. 571 #4A and 4B

Example 5 Solve a Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

Example 5 A.m  A = 36°, m  B = 54°, AB = 13.6 B.m  A = 54°, m  B = 36°, AB = 13.6 C.m  A = 36°, m  B = 54°, AB = 16.3 D.m  A = 54°, m  B = 36°, AB = 16.3 Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Angles of Elevation and Depression

Example 1 Angle of Elevation CIRCUS ACTS At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6- inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°? Make a drawing.

Example 2 Angle of Depression DISTANCE Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot? Make a sketch of the situation. Since are parallel, m  BAC = m  ACD by the Alternate Interior Angles Theorem.

Example 1 A.37° B.35° C.40° D.50° DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree?

Example 2 A.19 ft B.20 ft C.44 ft D.58 ft Luisa is in a hot air balloon 30 feet above the ground. She sees the landing spot at an angle of depression of 34 . What is the horizontal distance between the hot air balloon and the landing spot to the nearest foot?

Example 3 Use Two Angles of Elevation or Depression DISTANCE Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter.

Example 3 A.14 ft B.24 ft C.37 ft D.49 ft Madison looks out her second-floor window, which is 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window and the other car is parked directly across the street from the first car. The angles of depression of Madison’s line of sight to the cars are 17° and 31°. Find the distance between the two cars to the nearest foot.