Right Triangle Trig Review

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Right Triangle Trig Review
Given the right triangle from the origin to the
point (x, y) with the angle , we can find the
following trig functions
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Replacing (x, y) with these new values, we get
the point as
Moving to the circle centered at the origin
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Moving to the circle centered at the origin with
radius r, we find two points A and B.
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We can use the distance formula to find the
distance AB.
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Next, construct the angle in a circle with
the same radius r. Using the SAS property, the
triangle AOB in the previous example is congruent
to the triangle COD in this example. Therefore,
the length of segment AB must equal the length of
segment CD.
It must also be true that
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Finding points C and D and the length CD, we get
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By similar triangles, we know the length of AB
length of CD.
We can square both sides to get rid of the square
roots.
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Simplifying by squaring each group, we get
Every term has an r2. Divide each term by r2.
Using the pythagorean identity, we know
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Simplifying, we get
Subtracting the 2s from each side, we get
Each term has a -2, so divide out the -2.
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However, recall that Replacing in the
equation, we get
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To find a rule for , we
replace v with v.
Simplifying with odd/even rules, we get
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To get the sum/difference rules for sin, we will
use the co-function rule.
Lets use the cosine rule to find
Using the cosine sum rule
Using the co-function rules, we get
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Therefore
To get the sin(uv) rule,
Using the odd/even functions, we get