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Area of Triangle Using Trigonometry |
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We are all familiar with the formula for
the area of a triangle,
 ,
where b stands for the base and h stands for the height
drawn to that base. |
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In the triangle at the right, the
area could be expressed as:
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Now, let's be a bit more creative and
look at the diagram again. By using the right triangle on the left
side of the diagram, and
our knowledge of trigonometry, we can state that:
This tells us that the height,
h, can be expressed as bsinC.
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If we substitute this new
expression for the height, we can write the area formula as:
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We have just discovered that the area of a triangle
can be expressed using the lengths of two sides and the sine of the
included angle. This is often referred to as the SAS Formula
for the area of a triangle.
The "letters" in the formula may change from problem
to problem, so try to remember the pattern
"two sides and the sine of the included angle".
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"Wow! A trig area formula for triangles!!!" |
We no longer have to rely on a
problem supplying us with the length of the altitude (height) of the
triangle. If we know two sides and the included angle, we are
in business. |
Example 1:
Given the triangle at
the right, find its area. Express the area rounded to three
decimal places.
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Be careful!!!
When using your graphing calculator, be
sure that you are in DEGREE Mode, or that you are
using the degree symbol. |
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Example 2:
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square
units.
