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October 14th, 2010, 07:52 PM  #1 
Member Joined: Sep 2010 Posts: 37 Thanks: 0  trigonometry radians
Triangle drawn inside a circle using the diameter of the circle as one side with the other 2 sides meeting at a point on the circumference of the circle. If one of the angles of the triangle is 0.95 radians and the length of the adjacent side is 10units, what is the radius of the circle? How? Also, how many radians are there in a triangle? River running parallel to the base of a cliff has a bridge over it which is 4 m longer than the river is wide. From the top of a 60m high cliff the angle of depression is 0.71 radians for one end of the bridge and 1.08 radians for the other end. How wide is the river? I don't even get how to draw the diagram for that question. Thanks. 
October 14th, 2010, 08:45 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: trigonometry radians
Use the fact that the hypotenuse of a right triangle is a diameter of the triangle's circumcircle. The angle made where the shorter legs of the triangle meet on the circle is a right angle, meaning the third angle is complementary to the given angle of 0.95 radians, or ?/2  0.95. Then use the law of sines to state: sin(?/2)/2r = sin(?/2  0.95)/10 Use sin(?/2  x) = cos(x), and multiply through by 2. 1/r = cos(0.95)/5 Invert both sides. r = 5/cos(0.95) ? 8.596 The sum of the interior angles in a triangle is ? radians. 
October 14th, 2010, 09:32 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: trigonometry radians
[attachment=0vfwh5vu]tanhelp.JPG[/attachmentvfwh5vu] w = width of river, giving x + y = 4 m. tan(?/2  0.71) = (w + x)/60 tan(?/2  1.0 = (w + y)/60 Can you go from there? 
October 15th, 2010, 12:04 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: trigonometry radians
You can simplify further using the identity tan(?/2  ?) = cot(?), giving tan(0.71) = 60/(w + x) tan(1.0 = 60/(w + y) If you refer to the sketch, you'll see this makes sense. Using x + y = 4, we have [attachment=0:2471kkdd]tanhelp.JPG[/attachment:2471kkdd] w + x = 60/tan(0.71) w + 4  x = 60/tan(1.0 Adding the two equations, we have 2w + 4 = 60(cot(0.71) + cot(1.0) Dividing through by 2, then subtracting 2 from each side gives w = 30(cot(0.71) + cot(1.0)  2 w ? 48.94 m. 
October 20th, 2010, 04:22 AM  #5 
Member Joined: Sep 2010 Posts: 37 Thanks: 0  Re: trigonometry radians
Okay....But, the solution given is 33.7m

October 20th, 2010, 08:44 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,739 Thanks: 1000 Math Focus: Elementary mathematics and beyond  Re: trigonometry radians
60 * tan(?/2  0.71)  60 * tan(?/2  1.0  4 ? 33.74, where ?/2  0.71 is the angle from the base of the cliff to the outer edge of the bridge and ?/2  1.08 is the angle from the base of the cliff to the inner edge of the bridge.

October 20th, 2010, 09:27 AM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: trigonometry radians
Sorry, I assumed, and as usual, having a 50% chance, made the wrong assumption! 
October 21st, 2010, 01:46 AM  #8 
Member Joined: Sep 2010 Posts: 37 Thanks: 0  Re: trigonometry radians
Could you draw the diagram ? I'm a bit confused still. THanks 
October 21st, 2010, 06:47 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 18,586 Thanks: 1489 
You need to assume that the measurements are made from a position (on the clifftop) in the same vertical plane as the ends of the bridge, then use a diagram which shows two lines joining the ends of the bridge to a single point on the clifftop at a height of 60m above the level of the ends of the bridge. The ends of the bridge are then 60cot(0.71)m and 60cot(1.0m measured horizontally from the observation point, and so these values differ by the distance between the ends of the bridge. 

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