Download Right Angle Trigonometry Homework Solutions 1. The hypoteneuse

Right Angle Trigonometry Homework Solutions
1. The hypoteneuse of a right triangle has length 10. One of the angles is 32 degrees. Find the length
of the side opposite and also the length of the side adjacent to the angle of 34 degrees.
Answer(rounded to the nearest thousandth): Call a the side opposite. Then
a = 10 £ sin(32) = 5:299
Call b the side adjacent. Then
a
= sin(32o ) so
10
b
= cos(32o ) so b = 10 £ cos(32) = 8:480
10
2. A tree is 50 feet tall. You measure the angle of elevation from the ground where you are standing
to the top of the tree and get 50 degrees. How far away from the tree are you?
Answer: Call x the distance to the tree. Then
50
50
= tan(50) so x =
= 41:955 feet (rounded
x
tan(50)
to the nearest thousandth of a foot)
3. When you stand at a certain spot and measure from the ground where you are standing to the
top of a nearby building, you get an elevation of 70 degrees. You then walk 10 feet away from the
building and then you get the angle of elevation to be 50 degrees. How tall is the building?
Answer: Call x the distance from the ¯rst spot (where the angle of elevation is 70 degrees.) Call y
y
y
the height of the building. Then = tan(70) and
= tan(50)
x
x + 10
y = x £ tan(70) = (x + 10) £ tan(50). Solve for x and get x =
10 £ tan(50)
tan(70) ¡ tan(50)
Rounded to the nearest thousandth, x = 7:660, and y = x £ tan(70) = 21:047 is the height of the
building.
4. You are standing 20 feet from a building. The angle of elevation from the ground where you are
standing to the top of the building is 30 degrees. There is a billboard on the roof that is in line
with the side of the building facing you. The angle of elevation to the top of the billboard is 45
degrees. How tall is the billboard?
Answer: Call y the height of the building, and call y2 the height of the billboard. Since the angle to
y
the top of the billboard is 45 degrees, y +y2 = 20: We also know
= tan(30) so y = 20£tan(30) =
20
11:547 and y = 20 ¡ 11:547 = 8:453 to the nearest thousandth.
The sketches:
5. Find the value of x for each of these right triangles:
Answer: a) 2.77, b) 85.18, c) 62.76
6. A surveyor stands two hundred meters downstream from a bridge. The angle between the river
bank and the far end of the bridge is 30o . How wide is the river?
Answer: 115.47 m
7. A 40 foot pole is secured by cables which make an angle of 30o with the ground. How long is each
cable?
Answer: 80 feet
8. You are on top of a building (say, Wells Hall) looking at a smokestack which is 300 feet from the
building. The angle of elevation to the top of the smokestack is 20o , the angle of declension is 50o .
How tall is the smokestack?
Answer: 530.73 feet