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Trigonometry |
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Sum to product and
product to sum formulas or identities |
Sum to product formulas
for the tangent and the cotangent functions |
The product to sum
formulas for the sine and cosine functions |
Trigonometric identities,
examples |
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Sum to product and
product to sum formulas or identities |
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Sum to product formulas
for the tangent and the cotangent functions |
From the definition of the function tangent, |
![](T13B.gif) |
or |
![](T13C.gif) |
and |
![](T13D.gif) |
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and
for the function cotangent |
![](T13E.gif) |
or |
![](T13F.gif) |
and |
![](T13G.gif) |
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Using
the same method, |
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![](T13H.gif) |
and |
![](T13I.gif) |
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The product to sum
formulas for the sine and cosine functions |
By adding and subtracting addition formulas derived are following product
to sum formulas, |
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![](T14.gif) |
and |
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![](T14B.gif) |
and |
![](T14C.gif) |
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Trigonometric identities,
examples |
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Example:
Verify the identity |
![](TrigEx17.gif) |
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Solution:
We divide the numerator and
denominator on the left side by sin
a
and to the right side we use the cotangent formula for the
difference of two angles, thus
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![](TrigEx17A.gif) |
![](TrigEx17B.gif) |
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Example:
Express tan
3x in terms of tan
x. |
Solution:
Using the sum formula and the double angle formula for the
tangent function, |
![](TrigEx18.gif) |
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Example:
Prove the identity |
![](TrigEx19.gif) |
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Example:
Prove the identity |
![](TrigEx20.gif) |
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Example:
If tan
a
= 3/4, find tan
a/2.
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Solution:
Use formula |
![](TrigEx21.gif) |
to express tan
a/2
in terms of tan
a. |
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![](TrigEx21A.gif) |
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Example:
Prove the identity |
![](TrigEx22.gif) |
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Solution:
Substitute |
![](TrigEx22A.gif) |
then |
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![](TrigEx22B.gif) |
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Example:
Express the given
difference sin
61° -
sin 59° as a product.
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Solution:
Since |
![](TrigEx23.gif) |
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![](TrigEx23A.gif) |
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Example:
Prove the identity sin a
+ sin (a
+ 120°) +
sin (a
+ 240°)
= 0.
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Solution:
Applying the sum formula to
the last two terms on the left side of the identity we get,
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![](TrigEx24.gif) |
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Example:
Prove the identity |
![](TrigEx25.gif) |
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Solution:
Using the formula for the sum of the tangent |
![](TrigEx25A.gif) |
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![](TrigEx25B.gif) |
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Pre-calculus contents
F |
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