What relation gives the height of a ramp with a $4.76^circ$ incline, beginning $15$ feet from a building?
$begingroup$
In an attempt to study for an Algebra 2/Trigonometry Regent, (the Regents are standardized tests given out by the State of NY), I was using previous tests as practice tests. (All previous Regents could be found with the URL: http://www.nysedregents.org). I ran into a multiple choice question which I thought I answered correctly, but was actually a question I got wrong. I still think I got the question correct and I would love for anyone to enlighten me, (and help me study for an upcoming Regent), by telling me how I got the question incorrect... (The upcoming question is question 14 from an Algebra2/Trigonometry Regent from June of 2015).
"14.
By law, a wheelchair service ramp may be inclined no more than 4.76 degrees. If the base of the ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building's entrance?
(1) sin 4.76 = h/15
(2) sin 4.76 = 15/h
(3) tan 4.76 = h/15
(4) tan 4.76 = 15/h
The correct answer was option 3. This is understandable. If you would draw out the triangle, the Tangent of the angle, 4.76, is the opposite facing side over the angle's adjacent side; this would be h/15.
But I didn't choose option 3... I chose option 1. Reason being, if you decided to approach the question differently, using the Law of Sines instead of trying to figure out what the Tan of 4.76 is, you would get option 1.
The Law of Sines is as follows:
a/sin A = b/sin B.
In our problem we have to angles. We have the 4.76 angle, and we a 90 degree angle. We also have the length of one of the triangle's sides, 15, but we are missing the height, which is another of the triangle's sides. In order to figure out the length of the missing side, the height, we could use the Law of Sines:
15/sin 90 = h/sin 4.76
You cross multiply which would result in...
15 * (sin 4.76) = h * (sin 90)
Sin 90 = 1, so...
15 * (sin 4.76) = h * 1
or...
15 * (sin 4.76) = h
Divide by 15 on both sides...
sin 4.76 = h/15
This is option 1.
We have just established why option 1 is correct. But we also established why option 3 is correct. And, the correct answer was in fact option 3. Why? If we use the Law of Sines, shouldn't option 1 be correct as well?
trigonometry
edited Oct 28 '18 at 22:50
Blue
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
$endgroup$
add a comment |
$begingroup$
In an attempt to study for an Algebra 2/Trigonometry Regent, (the Regents are standardized tests given out by the State of NY), I was using previous tests as practice tests. (All previous Regents could be found with the URL: http://www.nysedregents.org). I ran into a multiple choice question which I thought I answered correctly, but was actually a question I got wrong. I still think I got the question correct and I would love for anyone to enlighten me, (and help me study for an upcoming Regent), by telling me how I got the question incorrect... (The upcoming question is question 14 from an Algebra2/Trigonometry Regent from June of 2015).
"14.
By law, a wheelchair service ramp may be inclined no more than 4.76 degrees. If the base of the ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building's entrance?
(1) sin 4.76 = h/15
(2) sin 4.76 = 15/h
(3) tan 4.76 = h/15
(4) tan 4.76 = 15/h
The correct answer was option 3. This is understandable. If you would draw out the triangle, the Tangent of the angle, 4.76, is the opposite facing side over the angle's adjacent side; this would be h/15.
But I didn't choose option 3... I chose option 1. Reason being, if you decided to approach the question differently, using the Law of Sines instead of trying to figure out what the Tan of 4.76 is, you would get option 1.
The Law of Sines is as follows:
a/sin A = b/sin B.
In our problem we have to angles. We have the 4.76 angle, and we a 90 degree angle. We also have the length of one of the triangle's sides, 15, but we are missing the height, which is another of the triangle's sides. In order to figure out the length of the missing side, the height, we could use the Law of Sines:
15/sin 90 = h/sin 4.76
You cross multiply which would result in...
15 * (sin 4.76) = h * (sin 90)
Sin 90 = 1, so...
15 * (sin 4.76) = h * 1
or...
15 * (sin 4.76) = h
Divide by 15 on both sides...
sin 4.76 = h/15
This is option 1.
We have just established why option 1 is correct. But we also established why option 3 is correct. And, the correct answer was in fact option 3. Why? If we use the Law of Sines, shouldn't option 1 be correct as well?
trigonometry
edited Oct 28 '18 at 22:50
Blue
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
$endgroup$
$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
3
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48
add a comment |
$begingroup$
In an attempt to study for an Algebra 2/Trigonometry Regent, (the Regents are standardized tests given out by the State of NY), I was using previous tests as practice tests. (All previous Regents could be found with the URL: http://www.nysedregents.org). I ran into a multiple choice question which I thought I answered correctly, but was actually a question I got wrong. I still think I got the question correct and I would love for anyone to enlighten me, (and help me study for an upcoming Regent), by telling me how I got the question incorrect... (The upcoming question is question 14 from an Algebra2/Trigonometry Regent from June of 2015).
"14.
By law, a wheelchair service ramp may be inclined no more than 4.76 degrees. If the base of the ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building's entrance?
(1) sin 4.76 = h/15
(2) sin 4.76 = 15/h
(3) tan 4.76 = h/15
(4) tan 4.76 = 15/h
The correct answer was option 3. This is understandable. If you would draw out the triangle, the Tangent of the angle, 4.76, is the opposite facing side over the angle's adjacent side; this would be h/15.
But I didn't choose option 3... I chose option 1. Reason being, if you decided to approach the question differently, using the Law of Sines instead of trying to figure out what the Tan of 4.76 is, you would get option 1.
The Law of Sines is as follows:
a/sin A = b/sin B.
In our problem we have to angles. We have the 4.76 angle, and we a 90 degree angle. We also have the length of one of the triangle's sides, 15, but we are missing the height, which is another of the triangle's sides. In order to figure out the length of the missing side, the height, we could use the Law of Sines:
15/sin 90 = h/sin 4.76
You cross multiply which would result in...
15 * (sin 4.76) = h * (sin 90)
Sin 90 = 1, so...
15 * (sin 4.76) = h * 1
or...
15 * (sin 4.76) = h
Divide by 15 on both sides...
sin 4.76 = h/15
This is option 1.
We have just established why option 1 is correct. But we also established why option 3 is correct. And, the correct answer was in fact option 3. Why? If we use the Law of Sines, shouldn't option 1 be correct as well?
trigonometry
edited Oct 28 '18 at 22:50
Blue
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
$endgroup$
In an attempt to study for an Algebra 2/Trigonometry Regent, (the Regents are standardized tests given out by the State of NY), I was using previous tests as practice tests. (All previous Regents could be found with the URL: http://www.nysedregents.org). I ran into a multiple choice question which I thought I answered correctly, but was actually a question I got wrong. I still think I got the question correct and I would love for anyone to enlighten me, (and help me study for an upcoming Regent), by telling me how I got the question incorrect... (The upcoming question is question 14 from an Algebra2/Trigonometry Regent from June of 2015).
"14.
By law, a wheelchair service ramp may be inclined no more than 4.76 degrees. If the base of the ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building's entrance?
(1) sin 4.76 = h/15
(2) sin 4.76 = 15/h
(3) tan 4.76 = h/15
(4) tan 4.76 = 15/h
The correct answer was option 3. This is understandable. If you would draw out the triangle, the Tangent of the angle, 4.76, is the opposite facing side over the angle's adjacent side; this would be h/15.
But I didn't choose option 3... I chose option 1. Reason being, if you decided to approach the question differently, using the Law of Sines instead of trying to figure out what the Tan of 4.76 is, you would get option 1.
The Law of Sines is as follows:
a/sin A = b/sin B.
In our problem we have to angles. We have the 4.76 angle, and we a 90 degree angle. We also have the length of one of the triangle's sides, 15, but we are missing the height, which is another of the triangle's sides. In order to figure out the length of the missing side, the height, we could use the Law of Sines:
15/sin 90 = h/sin 4.76
You cross multiply which would result in...
15 * (sin 4.76) = h * (sin 90)
Sin 90 = 1, so...
15 * (sin 4.76) = h * 1
or...
15 * (sin 4.76) = h
Divide by 15 on both sides...
sin 4.76 = h/15
This is option 1.
We have just established why option 1 is correct. But we also established why option 3 is correct. And, the correct answer was in fact option 3. Why? If we use the Law of Sines, shouldn't option 1 be correct as well?
trigonometry
trigonometry
edited Oct 28 '18 at 22:50
Blue
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
edited Oct 28 '18 at 22:50
Blue
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
edited Oct 28 '18 at 22:50
Blue
48.7k870156
edited Oct 28 '18 at 22:50
Blue
48.7k870156
edited Oct 28 '18 at 22:50
Blue
48.7k870156
48.7k870156
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
asked Aug 11 '16 at 4:17
user3814413user3814413
1135
1135
$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
3
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48
add a comment |
$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
3
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48
$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
3
3
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The angle opposite $15$ isn't the right angle, it's the triangle's "other" acute angle, which has measure $90^circ - 4.75^circ$. Therefore, applying the Law of Sines should get you to
$$frac{15}{sin(90^circ-4.75^circ)} = frac{h}{sin 4.75^circ} tag{1}$$
Then, since $sin(90^circ-theta) equiv costheta$, this becomes
$$frac{15}{cos 4.75^circ} = frac{h}{sin 4.75^circ} tag{2}$$
so that
$$frac{h}{15} = frac{sin 4.75^circ}{cos 4.75^circ} = tan 4.75^circ tag{3}$$
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
$endgroup$
add a comment |
$begingroup$
You have misinterpreted the law of sines. $A$ is the angle opposite to the side with length $a$. The side with length $15$ is not opposite to the right angle. Hence your initial equation is incorrect.
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
The angle opposite $15$ isn't the right angle, it's the triangle's "other" acute angle, which has measure $90^circ - 4.75^circ$. Therefore, applying the Law of Sines should get you to
$$frac{15}{sin(90^circ-4.75^circ)} = frac{h}{sin 4.75^circ} tag{1}$$
Then, since $sin(90^circ-theta) equiv costheta$, this becomes
$$frac{15}{cos 4.75^circ} = frac{h}{sin 4.75^circ} tag{2}$$
so that
$$frac{h}{15} = frac{sin 4.75^circ}{cos 4.75^circ} = tan 4.75^circ tag{3}$$
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
$endgroup$
add a comment |
$begingroup$
The angle opposite $15$ isn't the right angle, it's the triangle's "other" acute angle, which has measure $90^circ - 4.75^circ$. Therefore, applying the Law of Sines should get you to
$$frac{15}{sin(90^circ-4.75^circ)} = frac{h}{sin 4.75^circ} tag{1}$$
Then, since $sin(90^circ-theta) equiv costheta$, this becomes
$$frac{15}{cos 4.75^circ} = frac{h}{sin 4.75^circ} tag{2}$$
so that
$$frac{h}{15} = frac{sin 4.75^circ}{cos 4.75^circ} = tan 4.75^circ tag{3}$$
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
$endgroup$
add a comment |
$begingroup$
The angle opposite $15$ isn't the right angle, it's the triangle's "other" acute angle, which has measure $90^circ - 4.75^circ$. Therefore, applying the Law of Sines should get you to
$$frac{15}{sin(90^circ-4.75^circ)} = frac{h}{sin 4.75^circ} tag{1}$$
Then, since $sin(90^circ-theta) equiv costheta$, this becomes
$$frac{15}{cos 4.75^circ} = frac{h}{sin 4.75^circ} tag{2}$$
so that
$$frac{h}{15} = frac{sin 4.75^circ}{cos 4.75^circ} = tan 4.75^circ tag{3}$$
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
$endgroup$
The angle opposite $15$ isn't the right angle, it's the triangle's "other" acute angle, which has measure $90^circ - 4.75^circ$. Therefore, applying the Law of Sines should get you to
$$frac{15}{sin(90^circ-4.75^circ)} = frac{h}{sin 4.75^circ} tag{1}$$
Then, since $sin(90^circ-theta) equiv costheta$, this becomes
$$frac{15}{cos 4.75^circ} = frac{h}{sin 4.75^circ} tag{2}$$
so that
$$frac{h}{15} = frac{sin 4.75^circ}{cos 4.75^circ} = tan 4.75^circ tag{3}$$
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
answered Aug 11 '16 at 4:33
BlueBlue
48.7k870156
48.7k870156
add a comment |
add a comment |
$begingroup$
You have misinterpreted the law of sines. $A$ is the angle opposite to the side with length $a$. The side with length $15$ is not opposite to the right angle. Hence your initial equation is incorrect.
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
$endgroup$
add a comment |
$begingroup$
You have misinterpreted the law of sines. $A$ is the angle opposite to the side with length $a$. The side with length $15$ is not opposite to the right angle. Hence your initial equation is incorrect.
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
$endgroup$
add a comment |
$begingroup$
You have misinterpreted the law of sines. $A$ is the angle opposite to the side with length $a$. The side with length $15$ is not opposite to the right angle. Hence your initial equation is incorrect.
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
$endgroup$
You have misinterpreted the law of sines. $A$ is the angle opposite to the side with length $a$. The side with length $15$ is not opposite to the right angle. Hence your initial equation is incorrect.
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
edited Aug 21 '16 at 11:10
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
answered Aug 11 '16 at 4:38
Anonymous_originalAnonymous_original
748
748
add a comment |
add a comment |
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$begingroup$
what is NY ${}{}{}{}$?
$endgroup$
– Jorge Fernández Hidalgo
Aug 11 '16 at 4:22
3
$begingroup$
1) assumes that $15$ is the hypotenuse of the triangle, i.e. that distance is measured along the slope. Not a reasonable interpretation.
$endgroup$
– André Nicolas
Aug 11 '16 at 4:22
$begingroup$
The question states that "the base of the ramp begins 15 feet from the base of a public building." This means that the horizontal distance is 15, and not the length of the ramp. (Curiously enough, since $4.76^{circ}$ is a rather small angle, if you actually solve for $h$ through the equations in (1) and (3), you end up with fairly similar answers.)
$endgroup$
– Joey Zou
Aug 11 '16 at 4:31
$begingroup$
Perhaps NY is a typo, and was intended to refer to "the knights who say Ni". youtu.be/zIV4poUZAQo
$endgroup$
– John Joy
Aug 14 '16 at 11:48