7.3 Right Triangle Trig Word Problems Precalculus 7.3 Right Triangle Trig Word Problems

Special Angle Names Angle of Elevation Angle of Depression From Horizontal Up Angle of Depression From Horizontal Down

Angle of Elevation and Depression Imagine you are standing here. The angle of elevation is measured from the horizontal up to the object.

Angle of Elevation and Depression The angle of depression is measured from the horizontal down to the object. Constructing a right triangle, we are able to use trig to solve the triangle.

Examples

Lighthouse & Sailboat Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? x 5.7o 150 ft. 150 ft. Construct a triangle and label the known parts. Use a variable for the unknown value.

Lighthouse & Sailboat Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? x 5.7o 150 ft. Set up an equation and solve.

Lighthouse & Sailboat x 5.7o 150 ft. Remember to use degree mode! x is approximately 1,503 ft.

River Width A surveyor is measuring a river’s width. He uses a tree and a big rock that are on the edge of the river on opposite sides. After turning through an angle of 90° at the big rock, he walks 100 meters away to his tent. He finds the angle from his walking path to the tree on the opposite side to be 25°. What is the width of the river? Draw a diagram to describe this situation. Label the variable(s)

River Width We are looking at the “opposite” and the “adjacent” from the given angle, so we will use tangent Multiply by 100 on both sides

Subway The DuPont Circle Metrorail Station in Washington DC has an escalator which carries passengers from the underground tunnel to the street above. If the angle of elevation of the escalator is 52° and a passenger rides the escalator for 188 ft, find the vertical distance between the tunnel and the street. In other words, how far below street level is the tunnel?

Subway We are looking at the “opposite” and the “hypotenuse” from the given angle so we will use sine Multiply by 188 on each side

Building Height A spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire? Construct the required triangles and label. 38o 35o 500 ft.

Building Height Write an equation and solve. Total height (t) = building height (b) + spire height (s) Solve for the spire height. s t Total Height b 38o 35o 500 ft.

Building Height Write an equation and solve. Building Height s t b 38o 500 ft.

Building Height Write an equation and solve. Total height (t) = building height (b) + spire height (s) s t b The height of the spire is approximately 41 feet. 38o 35o 500 ft.

Mountain Height A hiker measures the angle of elevation to a mountain peak in the distance at 28o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29o. How much higher is the mountain peak than the hiker? Construct a diagram and label. 1st measurement 28o. 2nd measurement 1,500 ft closer is 29o.

Mountain Height Adding labels to the diagram, we need to find h. h ft x ft Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28o is 1500 + x.

Mountain Height Now we have two equations with two variables. Solve by substitution. Solve each equation for h. Substitute.

Mountain Height Solve for x. Distribute. Get the x’s on one side and factor out the x. Divide.

Mountain Height However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height. The height of the mountain above the hiker is 19,562 ft.