Name: ______Date: ______
Geometry A U5D5 More Trig ApsWarm Up
1) Solve for the missing angles a) cos(x) = 0.5431 x ≈______
e) 7sin(θ) = 3 θ≈______f) 13cos(A) = 4 A ≈______
2) Draw and label a diagram of the path of an airplane climbing at an angle of 11° with the ground. Find, to the nearest foot, the ground distance the airplane has traveled when it has attained an altitude of 400 feet.
3) Solve for x & y4) Solve for x & y
5) The modern building shown below is built with an outer wall (shown on the left) that is not at a degree angle with the floor. The wall on the right is perpendicular to both the floor and ceiling.
What is the length of the slanted outer wall, ? What is the length of the main floor, ?
Geometry A U5D5 More Trig ApsDepression & Elevation
1) Indirect measurement.When we cannot measure things directly, we can use trigonometry. A person walks out from the base of a tree 40 feet to point C. The measure of the angle formed from the ground to the top of the tree is 35°. Find the height of the tree.
You can use right triangles to find distances, if you know an angle of elevation or an angle of depression. The figure below shows each of these kinds of angles.
The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. For example, if you are standing on the ground looking up at the top of a mountain, you could measure the angle of elevation.
The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. For example, if you were standing on top of a hill or a building, looking down at an object, you could measure the angle of depression.
You can measure these angles using a clinometer or a theodolite. People tend to use clinometers or theodolites to measure the height of trees and other tall objects. Here we will solve several problems involving these angles and distances.
2) How tall is the tree? You are 5 feet tall and standing away from a tree, and you measure the angle of elevation to be . How tall is the tree?
3) You are standing on top of a building, looking at park in the distance. The angle of depression is . If the building you are standing on is tall, how far away is the park?
Name: ______Date: ______
Geometry A U5D5 Sim Figs & Trig ApsWork Together
1) If the sides of 2 similar triangles are in the ratio of 3:5, then the ratio of their areas would be
1)8:52) 3:53) 9:254) 27:125
2) If the sides of 2 similar triangles are in the ratio of 3:5, then the ratio of their perimeters would be
1)8:52) 3:53) 9:254) 27:125
3) If the heights of 2 similar cones are in the ratio of 1:3, then the ratio of their volumes would be
1)1:32) 1:273) 1:94) 3:27
4)
1)242) 183) 114) 9.375
5) Using the diagram below where DE is a midsegment, if the perimeter of triangle ABC is 60 cm and AB= 20 cm; DE =14, find the length of AC.
1)12 cm
2)26 cm
3)33 cm
4)94 cm
6) In the diagram DE is a midsegment, if the , what is
1)125o
2)55o
3)45o
4)35o
7) In the diagram below, the vertices of ΔDEF are the midpoints of the sides of triangle ABC, and the perimeter of ΔABC is 36 cm. What is the perimeter of ΔDEF?
1)6
2)9
3)18
4) 72
8) In the diagram below of ΔACD, E is a point on and B is a point on , such that || . If AE = 3, ED = 6, and DC = 15, find the length of .
9) The areas of two rectangles are in a ratio of 25:81. If the smaller side of the larger rectangle is 36 find the length of the smaller side of the smaller rectangle.
10) In the diagram below of ΔABC, is a mid-segment of ΔABC, DE = 4x – 8 and
AC= 80. Find the value of x.
11)
12) Find the number of degrees in angle y.
13) A person measuresthe distance of 100 feet from its base. From that point P we could then measure the angle required to sight the top. If that angle, called the angle of elevation, turned out to be 37°. Find the height of the pole.
14) tan(x) = 0.5431 x ≈______15) tan(B) = 3.5 B ≈______
16) Solve for x17) Solve for
18) Solve for side AC.
19) Solve for x & y
Setup the trigonometric ratio & solve for x in each triangle. Round to the nearest 10th.
20]/ 21]
22]
/ 23]
* Find the length of side :
Angles of Elevation and Depression
Right Triangles and Bearings
Other Applications of Right Triangles
In general, you can use trigonometry to solve any problem that involves right triangle. The next few examples show different situations in which a right triangle can be used to find a length or a distance.
Example 6: The wheelchair ramp
In lesson 4 we introduced the following situation: you are building a ramp so that people in wheelchairs can access a building. If the ramp must have a height of , and the angle of the ramp must be about , how long must the ramp be?
Given that we know the angle of the ramp and the length of the side opposite the angle, we can use the sine ratio to find the length of the ramp, which is the hypotenuse of the triangle:
This may seem like a long ramp, but in fact a ramp angle is what is required by the Americans with Disabilities Act (ADA). This explains why many ramps are comprised of several sections, or have turns. The additional distance is needed to make up for the small slope.
Right triangle trigonometry is also used for measuring distances that could not actually be measured. The next example shows a calculation of the distance between the moon and the sun. This calculation requires that we know the distance from the earth to the moon. In chapter 5 you will learn the Law of Sines, an equation that is necessary for the calculation of the distance from the earth to the moon. In the following example, we assume this distance, and use a right triangle to find the distance between the moon and the sun.
Example 7: The earth, moon, and sun create a right triangle during the first quarter moon. The distance from the earth to the moon is about . What is the distance between the sun and the moon?
Solution:
Let the distance between the sun and the moon. We can use the tangent function to find the value of :
Therefore the distance between the sun and the moon is much larger than the distance between the earth and the moon.
(Source: Trigonometry from Earth to the Stars.)
Additional Topics
Video: Introduction to Inverse Trigonometric Functions on the Graphing Calculator to Solve Right Triangels
Lesson Summary
In this lesson we have returned to the topic of right triangle trigonometry, to solve real world problems that involve right triangles. To find lengths or distances, we have used angles of elevation, angles of depression, angles resulting from bearings in navigation, and other real situations that give rise to right triangles. In later chapters, you will extend the work of this chapter: you will learn to find missing angles using trig ratios, and you will learn how to determine the angles and sides of non-right triangles.
Points to Consider
- In what kinds of situations do right triangles naturally arise?
- Are their right triangles that cannot be solved?
Trigonometry can solve problems at astronomical scale as well as earthly even problems at a molecular or atomic scale. Why is this true?
Review Questions
- Solve the triangle
- Two friends are writing practice problems to study for a trigonometry test. Sam writes the following problem for his friend Anna to solve:
In right triangle , the measure of angle is , and the length of side is . Solve the triangle.
Anna tells Sam that the triangle cannot be solved. Sam says that she is wrong. Who is right? Explain your thinking.
3. Use the Pythagorean Theorem to verify the sides of the triangle in example 2.
4. Estimate the measure of angle in the triangle below using the fact that and . Use a calculator to find sine values. Estimate to the nearest degree.
5. The angle of elevation from the ground to the top of a flagpole is measured to be . If the measurement was taken from away, how tall is the flagpole?
6. From the top of a hill, the angle of depression to a house is measured to be . If the hill is tall, how far away is the house?
7. An airplane departs city and travels at a bearing of . City is directly south of city . When the plane is east of city , how far has the plan traveled? How far apart are city and City ?
9. A surveyor is measuring the width of a pond. She chooses a landmark on the opposite side of the pond, and measures the angle to this landmark from a point away from the original point. How wide is the pond?