8.3 Trigonometry

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8.3 Trigonometry
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• Similar right triangles have equivalent ratios
  for their corresponding sides. These equivalent
  ratios are called Trigonometric Ratios. For
  example if we have two triangles that have the
  following angles of 90, 36.86, and and 53.14
  degrees. The side lengths compared to one
  another will always be in the same proportion,
  that means as long as the two triangles have all
  3 angles congruent, their sides will obviously be
  different but if you take the legs of one
  triangle and divide them, the quotient you get
  back will be identical to the legs of the second
  triangle divided in the same order.

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• These ratios exist for all right triangles that
  are similar. The ratios have specific names.
  They are sine, cosine, and tangent. The names of
  the ratios identify which side lengths you are
  comparing. There are 3 other ratios and they are
  referred to as the reciprocal trig ratios,
  their values are found by taking the reciprocals
  of sine, cosine, and tangent.
• The reciprocal ratio of sine (sin) is cosecant
  (csc)
• The reciprocal ratio of cosine (cos) is secant
  (sec)
• The reciprocal ratio of tangent (tan) is
  cotangent (cot)

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• How to determine a trig ratio. First you have to
  be provided with an angle. We then label the
  sides of the triangle in relation to that angle
  (hypotenuse, opposite, and adjacent)

A
opposite
HYPOTENUSE
adjacent
B
C
If we are using angle A, then we label BC as the
opposite side, and AB as the adjacent side.
If we are using angle C, then we label AB as the
opposite side, and BC as the adjacent side.
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So when we talk about the sin(A) we are
indicating the ratio of the opposite leg divided
by the hypotenuse (when you do the division the
value will show as a decimal on your calculator),
that ratio will be exactly the same for all right
triangles that have an angle congruent to this
triangles angle A.
SOH-CAH-TOA
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On your own complete the following question.
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• Once we understand how to identify particular
  ratios we can use the ratios to find a particular
  distance or a particular angle measurement.

We must use our calculators for this, our
calculators have stored in them the ratios of the
infinitely many similar triangles that could be
created. For example, your calculator has been
programmed so that it knows the 6 different
ratios that can be created from a triangle having
angle measurements of 90, 89, and 1 degrees. It
then knows the 6 different ratios that can be
created from a triangle having 90, 88, and 2
degrees. And 90, 87, and 3 degrees. All the way
up to 90, 45, and 45. But it also has the
capabilities of knowing the 6 ratios of a
triangle of 90, 40.23, and 49.77, and all
triangles with real number angle measurements.
We will use our calculator to find a missing side
length first. When finding a missing
distance/length we must know the numerical value
of one of the triangles acute angles.
Now that we know which sides we have in relation
to the known angle, we can now use what trig
ratio?
opposite
hypotenuse
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Now we know what w is, we can now solve the
triangle (find all 6 parts)
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These two triangles are similar, based on their
angles we have AA. Thus regardless of what we
find their side lengths to be, we will always see
that the ratios of opposite/hypotenuse,
opposite/adjacent, and adjacent/hypotenuse for
both triangles will always be equal. You
provide me with any 2 values for either a leg or
hypotenuse and we will show that the ratios are
always the same.
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Use trig ratios to find missing angle
measurements.

• As long as you know two sides of a right triangle
  you should always be able to find an angle
  measurement using one of the 3 trig ratios.
• On your calculators you will have to use the
  sin-1 or cos-1 or tan-1. These are referred to
  as the inverse trig functions. Which we will
  type in a ratio of the side lengths and the
  calculator will spit back an angle measurement
  that always goes with that ratio.

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• Many real life applications use trigonometry.
  One of the applications we will investigate the
  most is indirect measurement. Indirect
  measurement is the process of determining a
  measurement without physically measuring it.

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In a right triangle, the legs are 6 and .
What is the length of the hypotenuse and what are
the measures of the two non-right (acute) angles?
Can this be done using different methods?
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