7052
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1
2011-02-01T00:00:01
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<p> </p>
<div>
<mdo:image alt="" src="diagram.png" style="width: 358px; height: 279px; float: left;" />In this diagram OA is a radius of a unit circle. The hypotenuse of the large triangle is tangent to the circle at A.</div>
<div> </div>
<div>Find the lengths $\cos(a)$, $\sin(a)$, $\tan(a)$, $\frac{1}{\cos(a)}$, $\frac{1}{\sin(a)}$  and $\frac{1}{\tan(a)}$ in the diagram.</div>
<div> </div>
<div>Find the areas of all of the regions in the diagram.</div>
<div> </div>
<div> </div>
<div> </div>
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<div class="framework" style="float:left; clear:left">
<span style="font-style: italic;">Did you know ... ?</span>
<br />
<br />
Whilst trigonometric functions are defined algebraically in more advanced applications, geometric images such as this one can give great insight into the relationships between the functions. They also impart a sense of the beauty and interconnectedness of mathematics, which inspires many students of mathematics.</div>
<div> </div>
<p>
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</p>
</mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
There are many ways to create the solution as it is something like
a jigsaw. We used this method<br></br>
<br></br>
1. Mark the right angles (use the fact the a radius and tangent at
a point are at right angles)<br></br>
2. Make all occurrences of the angle a<br></br>
3. Mark the unit length<br></br>
<br></br>
The diagram then becomes<br></br>
<br></br>
<mdo:image height="242" width="395" alt="" src="diagramSolution.png"></mdo:image><br></br>
To work out all of the areas we need to decide what unit of
measurement the angle $a$ is in. Of course, we choose radians:
there are $2\pi$ radians in a circle. We will also need to know the
formula for the area of a circle and the area of a triangle. In
this case, the areas are given as<br></br>
<br></br>
<mdo:image height="386" width="626" alt="" src="diagramSolution2.png"></mdo:image><br></br>
The two largest areas are equal for around 0.40523 radians (23.2
degrees).<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
This problem uses basic notions of trigonometry and areas of
triangles which are fundamental to developments in A-level
mathematics.<br></br>
<br></br>
Being able to work with diagrams such as this is useful prepation
for both coordinate geometry and mechanics.<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
There are many ways to create the solution as it is something like
a jigsaw. We used this method<br></br>
<br></br>
1. Mark the right angles (use the fact the a radius and tangent at
a point are at right angles)<br></br>
2. Make all occurrences of the angle a<br></br>
3. Mark the unit length<br></br>
<br></br>
The diagram then becomes<br></br>
<br></br>
<mdo:image height="242" width="395" alt="" src="diagramSolution.png"></mdo:image><br></br>
To work out all of the areas we need to decide what unit of
measurement the angle $a$ is in. Of course, we choose radians:
there are $2\pi$ radians in a circle. We will also need to know the
formula for the area of a circle and the area of a triangle. In
this case, the areas are given as<br></br>
<br></br>
<mdo:image height="386" width="626" alt="" src="diagramSolution2.png"></mdo:image><br></br>
The two largest areas are equal for around 0.40523 radians (23.2
degrees).<br></br>
<br></br></mdoxml>
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Weekly Challenge 3: Geometric Trig
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