|
Trigonometry |
|
Sum to product and
product to sum formulas or identities |
Sum to product formulas
for the sine and the cosine functions |
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 |
|
|
|
|
|
|
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,
|
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 |
|
|
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 |
|
|
|
thus, |
|
and |
|
|
|
|
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) |
|
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, |
|
|
|
thus, |
|
and |
|
|
|
|
Sum to product formulas
for the tangent and the cotangent functions |
From the definition of the function tangent, |
|
or |
|
and |
|
|
|
and
for the function cotangent |
|
or |
|
and |
|
|
|
Using
the same method, |
|
|
and |
|
|
|
|
The product to sum
formulas for the sine and cosine functions |
By adding and subtracting addition formulas derived are following product
to sum formulas, |
|
|
and |
|
|
|
|
|
and |
|
|
|
|
|
Trigonometric
identities
examples |
Example:
Prove the identity |
|
|
|
|
Example:
If tan
a
= 3/4, find tan
a/2.
|
Solution:
Use formula |
|
to express tan
a/2
in terms of tan
a. |
|
|
|
Example:
Prove the identity |
|
|
Solution:
Substitute |
|
then |
|
|
|
Example:
Express the given
difference sin
61° -
sin 59° as a product.
|
Solution:
Since |
|
|
|
|
Example:
Prove the identity sin a
+ sin (a
+ 120°) +
sin (a
+ 240°)
= 0.
|
Solution:
Applying the sum formula to
the last two terms on the left side of the identity we get,
|
|
|
Example:
Prove the identity |
|
|
Solution:
Using the formula for the sum of the tangent |
|
|
|
|
Example:
Prove that |
|
|
Solution: Replace
sin a
by cos (p/2
-
a)
and cos a
by sin (p/2
-
a)
and use the sum to product
formula
|
|
|
|
|
|
|
|
|
|
|
Functions
contents D |
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |