Right triangles can be used to model and solve many applied problems.
While visiting Niagara Falls, a tourist walking toward Horseshoe Falls on a walkway next to Niagara Parkway notices the entrance to the Cave of the Winds attraction directly across the Niagara River. She continues walking for another 1000 ft and finds that the entrance is still visible but at approximately a angle to the walkway.
a) How many feet is she from the entrance to the Cave of the Winds?
b) What is the approximate width of the Niagara River at that point?
a) We know the side adjacent to the angle and want to find the hypotenuse. We can use the cosine function:
After walking 1000 ft, she is approximately 1556 ft from the entrance to the Cave of the Winds.
b) We know the side adjacent to the angle and want to find the opposite side. We can use the tangent function:
The width of the Niagara River is approximately 1192 ft at that point.
Now Try Exercise 21.
House framers can use trigonometric functions to determine the lengths of rafters for a house. They first choose the pitch of the roof, or the ratio of the rise over the run. Then using a triangle with that ratio, they calculate the length of the rafter needed for the house. José is constructing rafters for a roof with a pitch on a house that is 42 ft wide. Find the length x of the rafter of the house to the nearest tenth of a foot.
We first find the angle that the rafter makes with the side wall. We know the rise, 10, and the run, 12, so we can use the tangent function to determine the angle that corresponds to the pitch of :
Using a calculator, we find that . Since trigonometric function values of depend only on the measure of the angle and not on the size of the triangle, the angle for the rafter is also .
To determine the length x of the rafter, we can use the cosine function. (See the figure at left.) Note that the width of the house is 42 ft, and a leg of this triangle is half that length, 21 ft.
The length of the rafter for this house is approximately 27.3 ft.
Now Try Exercise 33.
Many applications with right triangles involve an angle of elevation or an angle of depression. The angle between the horizontal and a line of sight above the horizontal is called an angle of elevation. The angle between the horizontal and a line of sight below the horizontal is called an angle of depression. For example, suppose that you are looking straight ahead and then you move your eyes up to look at an approaching airplane. The angle that your eyes pass through is an angle of elevation. If the pilot of the plane is looking forward and then looks down, the pilot’s eyes pass through an angle of depression.
In Telluride, Colorado, there is a free gondola ride that provides a spectacular view of the town and the surrounding mountains. The gondolas that begin in the town at an elevation of 8725 ft travel 5750 ft to Station St. Sophia, whose altitude is 10,550 ft. They then continue 3913 ft to Mountain Village, whose elevation is 9500 ft.
a) What is the angle of elevation from the town to Station St. Sophia?
b) What is the angle of depression from Station St. Sophia to Mountain Village?
We begin by labeling a drawing with the given information.
a) The difference in the elevation of Station St. Sophia and the elevation of the town is , or 1825 ft. This measure is the length of the side opposite the angle of elevation, , in the right triangle shown at left. Since we know the side opposite and the hypotenuse, we can find by using the sine function. We first find :
Using a calculator, we find that
Thus the angle of elevation from the town to Station St. Sophia is approximately .
b) When parallel lines are cut by a transversal, alternate interior angles are equal. Thus the angle of depression, . from Station St. Sophia to Mountain Village is equal to the angle of elevation from Mountain Village to Station St. Sophia, so we can use the right triangle shown at left.
The difference in the elevation of Station St. Sophia and the elevation of Mountain Village is , or 1050 ft. Since we know the side opposite the angle of elevation and the hypotenuse, we can again use the sine function:
Using a calculator, we find that
The angle of depression from Station St. Sophia to Mountain Village is approximately .
Now Try Exercise 17.
Bamboo is the fastest growing land plant in the world and is becoming a popular wood for hardwood flooring. It can grow up to 46 in. per day and reaches its maximum height and girth in one season of growth. (Sources: Farm Show, Vol. 34, No. 4, 2010, p. 7; U-Cut Bamboo Business; American Bamboo Society) To estimate the height of a bamboo shoot, a farmer walks off 27 ft from the base and estimates the angle of elevation to the top of the shoot to be . What is the approximate height h of the bamboo shoot?
From the figure, we have
The height of the bamboo shoot is approximately 74 ft.
Some applications of trigonometry involve the concept of direction, or bearing. In this text, we present two ways of giving direction, the first below and the second in Section 6.3.