Download AAG ACDM - Unit 8 Part 2 Applications and Word Problems

Name:______________________________________Date:_____________Pd:__________
A.A.G. ACADEMIC
Unit 8 Part 2 – Applications & Solving Trigonometric Ratios
Finding Lengths and Angles Using the Three Trig Relationships (Sine, Cosine, & Tangent)
Situation 1: When the unknown is in the Numerator of the ratio.
Gertie the gopher is looking at a tree that is 100 feet away from her nest in the horizontal direction. If the
angle of elevation between Gertie’s nest and the top of the tree is 13°, then how tall is the tree?
Situation 2: When the unknown is the Denominator of the ratio.
A plane is 30,000 feet off of the ground when it begins its approach to the runway. In the horizontal
direction, how far is the plane from the runway if the pilot lowers the nose at an angle of depression of 4.5o
to meet the runway?
Situation 3: When the unknown is the Angle Measure in the equation.
Finding Angle Measures Using Trigonometric Function Inverses
If you are given the sine, cosine or tangent of an angle, use their inverse
functions: sin-1, cos-1, or tan-1 to find the corresponding angle. Use a
calculator to find  to the nearest degree.
A)
tan  = 0.25
B)
sin  = 0.61
C)
cos  = 0.80
Page | 1
Review of Angles of Depression and Elevation
Horizontal
(High Level)
Angle of Depression
Line of Sight
Angle of Elevation
Horizontal
(Low Level)
For all questions without a diagram, you are required to draw and label your own. Please show all work
and calculations and round to the nearest hundredth when necessary. Be sure to label all units of
measurement.
1. A soldier in a 75-foot tower spots a fellow soldier’s vehicle approaching in the distance. The
soldier in the tower spots the vehicle at an angle of depression of 8o. From the ground, how far is
the vehicle from the tower (the horizontal distance)?
2. A ski lift is attached to a wire from the base of the mountain to the peak. The wire on the lift is
3,500 feet long and is attached directly from the ground to the peak with an angle of elevation to the
top of 25o. Assuming this situation forms a right triangle, what is the height of the mountain?
3. A plane flying at an altitude of 33,000 feet is 130 miles away from the airport when it begins its
descent. If the angle of descent (depression) is constant, then find this angle measure. Think of
how to find angle measure and be careful of the units!
Page | 2
4. How far up on the side of a building can a 15-m ladder reach if the measure of the angle it makes
with the ground may not exceed 72? You must draw a diagram.
5.
6. A ship sails 64 kilometers on a bearing of 20°. A bearing, is the angle traveled away from ‘Due
North’. How far east of its original position is the ship?
7. A straight water slide makes a 40° angle with the surface of the water. If the slide is 11.5 meters
high, how long is it? You must draw a diagram.
Page | 3
8.
9. When the sun is at an angle of elevation 25o up from the horizon, a tree casts a shadow 14.5 feet
long. How tall is the tree?
______________________
10. A 30-foot extension ladder is safe if the angle it makes with the ground is between 60o and 75o .
a. What is the farthest up on a vertical wall that a 30-foot ladder of this type can reach up and still
be safe?
b. How far, at a minimum, should the foot of the ladder be placed from the base of the wall?
Page | 4
11. A Destroyer fires a projectile with a range of 13 miles at a bearing of 30 east of due north.
Remember: A bearing is an angle measured clockwise from due north. Draw a “Birds-Eye” view of
the situation like #6. Find:
a. how far north will the projectile land?
b. how far east will the projectile land?
12. Find the measures of the acute angles in a 5-12-13 right triangle. Draw and label a diagram.
13. A ramp is built that leads up to a doorway. Its slope is 1/13. Remember that slope is Rise/Run;
find the angle that the ramp will make with the horizontal?
14. To ensure that water and waste are not trapped in a drainpipe, drainpipes are installed so that every
8 feet of pipe, there is a drop of 1 inch. What angle does the pipe make with the horizontal?
Page | 5
15. An airplane is flying in a horizontal straight line towards an airport. Find its altitude at the moment
when the angle of depression is 6 and the plane is 100 miles away from its destination. Picture
required.
16. Consider the picture at the right.
a. Name all angles of elevation.
b. True or False: m2 = m5
c. Name all angles of depression.
d. Name two pairs of congruent angles.
17. An observer on a sea cliff with a height of 12m spots a shark-fin through a pair of binoculars at an
angle of depression of 5.7.
a. To the nearest meter, how far is the shark from the base of the cliff?
b. A few minutes later, the observer spots the same shark at an angle of depression of 7.6. To the
nearest meter, how much closer has the shark moved to the base of the cliff?
Page | 6
18. The angles of depression to the near and far banks of a river measure 49 and 11, respectively. If
the observer’s eyes are 1.8m above the ground, how wide is the river? Be sure to draw a diagram.
19. Find the area of a Regular Decagon with sides of 15 yards.
Page | 7
20. Find the Area and Perimeter of Triangle ABC below. Be sure to show and neatly organize your
work.
21. Find the measure of the smallest angle in each of the Pythagorean Triples below. You should
draw/label a diagram and write/solve an equation to show your work.
a.) 8, 15, and 17
b.) 9, 40, and 41
22. Find the approximate perimeter and area of a right triangle that also has an angle measuring 12o that
forms a side length of 8 cm (opposite side). Draw and label a diagram.
Page | 8