Q. For example, i know that the latitude is 30 degrees N. and the noon sun angle is 60 degrees on March 21; however i do not know how to calculate the length of day in hours given this information.
A. Consider the sun's position as a great circle in the sky. You know that a full circle take 24 hours. Given the maximum angle, how much of the circle's circumference is exposed above the horizon? This is a problem involving a chord of the circle. Can you now picture enough to solve the problem? You're given the elevation from the midpoint of the chord to the farthest point of the nearer arc; what angle does that arc subtend from the center of the circle?
A convex octagon is inscribed in a circle and has four consecutive sides of length 3 and four consecutive?
Q. A convex octagon is inscribed in a circle and has four consecutive sides of length 3 and four consecutive sides of length 2. Find its area in the form r+s*sqrt(t), where r,s, and t are integers.
A. Suppose that 2a is the measure of the arc whose chord is length 2, and 2b is the measure of the arc whose chord is length 3. Suppose r is the radius of the circle. We have
.. r*sin(a) = 2/2
.. r*sin(b) = 3/2
We also know that 2a+2b = 90°, so a+b = 45°. (It makes no difference whether the sides of the same measure are consecutive. They can be rearranged so they alternate with sides of the other measure.)
We can use these equations to calculate the cotangent of the angle a.
.. sin(45°-a)/sin(a) = 3/2 = (cos(a)/â2 - sin(a)/â2)/sin(a) = cot(a)/â2 - 1/â2
.. cot(a) = (3+â2)/â2
From above, we know that
.. r = csc(a) = â(1+cot(a)^2) = â(13/2 + 3â2)
Any of a variety of methods can be used to find the area given r. One I tried was Heron's formula for the area of a triangle given its side lengths. This gives
.. (octagon area) = 4â(r^2-1) + 3â(4r^2-9)
Substituting the above value of r into this equation, we get
.. (octagon area) = 4â(11/2 + 3â2) + 3â(17 + 12â2)
I used a math tool (see source list) to simplify this to
.. (octagon area) = 13 + 12â2
So, {r, s, t} = {13, 12, 2}.
.. r*sin(a) = 2/2
.. r*sin(b) = 3/2
We also know that 2a+2b = 90°, so a+b = 45°. (It makes no difference whether the sides of the same measure are consecutive. They can be rearranged so they alternate with sides of the other measure.)
We can use these equations to calculate the cotangent of the angle a.
.. sin(45°-a)/sin(a) = 3/2 = (cos(a)/â2 - sin(a)/â2)/sin(a) = cot(a)/â2 - 1/â2
.. cot(a) = (3+â2)/â2
From above, we know that
.. r = csc(a) = â(1+cot(a)^2) = â(13/2 + 3â2)
Any of a variety of methods can be used to find the area given r. One I tried was Heron's formula for the area of a triangle given its side lengths. This gives
.. (octagon area) = 4â(r^2-1) + 3â(4r^2-9)
Substituting the above value of r into this equation, we get
.. (octagon area) = 4â(11/2 + 3â2) + 3â(17 + 12â2)
I used a math tool (see source list) to simplify this to
.. (octagon area) = 13 + 12â2
So, {r, s, t} = {13, 12, 2}.
How to calculate a section of a circle?
Q. I have two variable lines crossing each other at a point. I need to draw a section of a circle of known diameter so that it rounds the intersection. I can't figure out the math and trigonometry needed. Could you help? No, it is not a homework, I am 63 years old :-) and I try to trace with Flash ActionScript the route of a ship turning at a given turn rate on a predefined route. I need to calculate the start and end points where the circle is tangent to the two lines. Thanks in advance.
A. I don't think I've understood properly what you are trying to ask. I guess you want to calculate the length of the arc of a circle of known radius(say R) between the points where these two lines(passing through your fixed point, i.e., P) touch it. If that is so, then you just need to calculate the angle that these lines make at the point P, say that angle is found to be $. Since tangents make an angle of 90 degrees with the line joining centre to the point of contact, and the sum of angles of a quadrilateral(here we will consider the quadrilateral having vertices the two points of contact of these tangent lines, the centre of the circle, and the point P) is 360 degrees, the angle subtended by that arc(whose length we are calculating) on the centre of the circle is (180 - $) degrees. Since circumference of the whole circle is 2.pi.R, the length of this arc will be 2.pi.R.(180 - $)/360
But I have a feeling that I misunderstood your question. If that is the case please edit, and clarify.
But I have a feeling that I misunderstood your question. If that is the case please edit, and clarify.
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