1. A 16 foot ladder is leaning against a house.
It touches the bottom of a window that is 12 feet 6 inches above the ground.
What is the measure of the angle that the ladder forms with the ground?1. A 16 foot ladder is leaning against a house.
It touches the bottom of a window that is 12 feet 6 inches above the ground.
What is the measure of the angle that the ladder forms with the ground?
Let x equal the measure of the angle the ladder forms with the ground. A picture of the problem is drawn to the right.
We have the side opposite to the angle in question as well as the hypotenuse. I can write the unknown in terms of the known using the definition of sine:
. First I need to get every thing in terms of inches:
12 ft = 144 in. ,so 12 ft 6 in = 150 in. and 
= 192 in
so 
. Notice the units cancel out. You should always
get a unitless number when you have a trigonometric ratio. Using the inverse
sin on a calculator I get the measure of the angle is equal to 
.
2.
a. I would use the converse of the Pythagorean Theorem to solve this problem.
b. The converse of the Pythagorean Theorem tells me that if 
then the pole would be at a right angle with the ground when the string
was 17 ft.
So compute 
and 
since 
the pole would be at
a right angle with the ground when the string is 17 ft.
3. Kaila is flying a kite whose string is making a 
angle
with the ground. The kite string is 65 meters long. How far is the kite
above the ground?
After reading the problem I would draw the following picture:
where h is the height, what I want to find. I know an angle, so I know I need to use a trigonometric ratio to solve this problem. I am looking for the side opposite the given angle and I know the hypotenuse. Looking at my definitions I see I should use the sine ratio to write the unknown in terms of knowns.
So I have
or
.
So using my calculator to compute sin I find the
kite is approximately 61 meters above the ground.
4. The Brook's are installing a wide-screen television with a 60-inch diagonal. Their entertainment center is 48 inches wide by 36 inches high, will the television fit in their current entertainment center?
After reading the problem I draw the following:
I want to find d to see if the television with a 60- inch diagonal will fit into a rectangle that is 48-inches by 36 inches. I can use the Pythagorean Theorem to put my knowns in terms of my unknown:
or
so the television will fit exactly.
5. A surveyor is 100 meters from the base of a dam. The angle of elevation
to the top of the dam measures 
. The surveyor's eye-level
is 1.73 meters above the ground. Find the height of the dam to the nearest
hundredth of a meter.
After reading the problem I draw:
So the top of the dam will be distance d + 1.73 m. I have and angle,
and the side adjacent to the angle and I am looking for the side opposite
the angle. Since 
I will use this trigonometric ratio to solve
for d.
I get: 
I am not quite done, I must add the distance the triangle is above the ground to get the height I am looking for. The height of the dam is 48.77 m + 1.73 m = 50.50 m.