Section 9-3 Angles of Elevation and Depression SPI 32F: determine the trigonometric ratio for a right triangle needed to solve a real-world problem given a diagram

Objectives:• Use angles of Elevation and Depression to solve problems

Sin = opposite leg hypotenuseCos = adjacent leg hypotenuse

Tan = opposite adjacent

Tan -1 = opposite adjacent

Sin -1 = opposite hypotenuse

Cos -1 = adjacent hypotenuse

Use if you have an angle measure

Use if you need to find angle measures

Angles of Elevation and Depression

Angle of Elevation:Person on the ground looks at anobject

Angle of Depression Person looks down at anobject

Why are the two angles congruent?

Transversal and parallel lines (alternate interior angles)

Describe 1 and 2 as they relate to the situation shown.

One side of the angle of depression is a horizontal line. 1 is the angle of depression from the airplane to the building.

One side of the angle of elevation is a horizontal line. 2 is the angle of elevation from the building to the airplane.

Angles of Elevation and Depression

A theodolite is an instrument for measuring both horizontal and vertical angles, as used in triangulation networks. It is a key tool in surveying and engineering work, but theodolites have been adapted for other specialized purposes in fields like metrology and rocket launch technology.

Theodolite

A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the building is 35°. How tall is the building?

Draw a diagram to represent the situation.

x = 200 • tan 35° Solve for x.

Use the tangent ratio.tan 35° = x 200

Use a calculator.200 35 140.041508

So x 140.

To find the height of the building, add the height of the Theodolite, which is 5 ft tall.

The building is about 140 ft + 5 ft, or 145 ft tall.

Angles of Elevation and Depression

An airplane flying 3500 ft above ground begins a 2°

descent to land at an airport. How many miles from the

airport is the airplane when it starts its descent?

Draw a diagram to represent the situation.

Use the sine ratio.sin 2° =3500

x

x = 3500 sin 2°

Solve for x.

3500 2 100287.9792 Use a calculator.

5280 18.993935 Divide by 5280 to convert feet to miles.

The airplane is about 19 mi from the airport when it starts its descent.

Angles of Elevation and Depression