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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 sine and the cosine 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 |
Sum to product formulas
for the sine and the cosine functions |
Adding the sum and difference formulas for the sine function,
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sin (a
+ b)
= sin a
· cos b
+ cosa
· sin b
(1) |
sin (a
-
b)
= sin a
· cos b
-
cosa
· sin b
(2) |
yields |
sin (a
+ b)
+
sin (a
-
b)
= 2sin a
· cos b |
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and
by subtracting the second from the first identity, |
sin (a
+ b)
-
sin (a
-
b)
= 2cosa
· sin b. |
Then,
substitute a
+ b
= x and a
-
b
= y . |
By
adding and subtracting these equalities we get |
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thus, |
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and |
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Using the same procedure for the cosine function, |
cos
(a
+ b)
= cos a
· cos b
-
sin a
· sin b
(1) |
cos
(a
-
b)
= cos a
· cos b
+ sin a
· sin b
(2) |
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by
adding (1)
+
(2) we get,
cos
(a
+ b)
+ cos
(a
-
b)
= 2cos a
· cos b |
and subtracting
(1)
-
(2)
cos
(a
+ b)
-
cos
(a
-
b)
= -2sin
a
· sin b |
substitute, a
+ b
= x and a
-
b
= y
so that, |
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thus, |
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and |
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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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and |
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and |
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Trigonometric identities,
examples |
Example:
Using known values, sin
60°
= Ö3/2
and sin 45°
= Ö2/2
evaluate sin 105°.
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Solution: Applying the sum formula for the sine function,
sin (a
+ b)
= sin a
· cos b
+ cosa
· sin b, |
therefore,
sin 105°
= sin (60°
+ 45°)
= sin 60° · cos 45° + cos60°
· sin 45° |
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Example:
Prove the identity |
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Solution:
Using the addition formula |
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Example:
Verify the identity |
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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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Example:
Express sin
3x in terms of sin
x. |
Solution:
Using the sum formula and the double angle formula for the sine function, |
sin
3x
= sin (2x + x)
= sin 2x · cos x + cos 2x · sin x
= 2sin x cos x · cos x + (cos2 x
-
sin2 x) · sin x |
= 2sin x · (1 -
sin2 x) + (1 -
2sin2 x) · sin x = 3sin x -
4sin3 x. |
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Example:
Prove the identity |
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Example:
Express the given
difference sin
61° -
sin 59° as a product.
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Solution:
Since |
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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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Example:
Prove that |
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Solution: Replace
sin a
by cos (p/2
-
a)
and cos a
by sin (p/2
-
a)
and use the sum to product
formula
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Pre-calculus contents
F |
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