Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.8 Part 1 Applications of Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Solve a right triangle. Solve problems involving bearings. Model simple harmonic motion.

Solving Right Triangles Solving a right triangle means finding the missing lengths of its sides and the measurements of its angles. We will label right triangles so that side a is opposite angle A, side b is opposite angle B, and side c, the hypotenuse, is opposite right angle C. When solving a right triangle, we will use the sine, cosine, and tangent functions, rather than their reciprocals.

Example: Solving a Right Triangle Let A = 62.7° and a = 8.4. Solve the right triangle, rounding lengths to two decimal places. Find the other acute angle B. Find one missing side. Side b using angle B and side a.

Example: Solving a Right Triangle - continued Find the other missing side. Side c using angle A and side a.

Example – Solving a Right Triangle Solve the right triangle shown for all unknown sides and angles. Solution: Because C = 90 , it follows that A + B = 90  and B = 90  – 34.2  = 55.8.

To solve for a, use the fact that a = b tan A. Example – Solution To solve for a, use the fact that a = b tan A. So, a = 19.4 tan 34.2   13.18. Similarly, to solve for c, use the fact that  23.46.

Your Turn – Solving a Right Triangle Find the remaining parts of the triangle.

Solving an Applied Trigonometry Problem Draw a sketch, and label it with the given information. Label the quantity to be with a variable. Use the sketch to write an equation relating the given quantities to the variable. Solve the equation, and check that your answer makes sense.

Angles of Elevation and Depression  =  Angle of Depression Angle of Elevation

Example: Finding a Side of a Right Triangle From a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is 85.4°. Approximate the height of the Eiffel Tower to the nearest foot. The height of the Eiffel Tower is approximately 994 feet.

Example – Finding a Side of a Right Triangle A safety regulation states that the maximum angle of elevation for a rescue ladder is 72. A fire department’s longest ladder is 110 feet. What is the maximum safe rescue height? Solution: A sketch is shown. From the equation sin A = a / c, it follows that a = c sin A Figure 4.79

Example 2 – Solution = 110 sin 72  104.6. cont’d = 110 sin 72  104.6. So, the maximum safe rescue height is about 104.6 feet above the height of the fire truck.

Your Turn: The angle of elevation of the Sun is 35.1 ̊ at the instant it casts a shadow 789 feet long of a water tower. Use this information to calculate the height of the water tower.

Your Turn: Bob is at the top of a vertical cliff, 80m high. He sees a boat out at sea. The angle of depression from Bob to the boat is 34o. How far from the cliff base is the boat?

Example: Finding an Angle of a Right Triangle A guy wire is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the ground. Find the angle, to the nearest tenth of a degree, that the wire makes with the ground. The wire makes an angle of approximately 29.0° with the ground.

Your Turn: A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown. Find the angle of depression of the bottom of the pool.

Your Turn: The length of the shadow of a tree 22.02 m tall is 28.34 m. Find the angle of elevation of the sun.

Example - Find of an Object in the Distance Finding the height of a tree on a mountain.

Example - Find of an Object in the Distance Finding the height of a tree on a mountain.

Example - Find of an Object in the Distance Finding the height of a tree on a mountain.

Your Turn: At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35 ̊, while the angle to the top is 53 ̊ . Find the height, s, of the smokestack alone.