|
| Trigonometry |
|
Trigonometric
functions of double angles, double angle formulas |
|
Trigonometric functions expressed by the half
angle |
|
Trigonometric identities,
examples |
|
|
|
|
|
|
|
| Trigonometric
functions of double angles, double angle formulas |
|
By substituting b
with
a
in the sum formulas, |
|
sin (a
+ b)
= sin a
· cos b
+ cosa
· sin b
and cos
(a
+ b)
= cos a
· cos b
-
sin a
· sin b, |
 |
and |
 |
|
| thus
for example, sin
2a
= sin (a
+ a)
= sin a
· cos a
+ cosa
· sin a
= 2sin a
cos a
so we get, |
| |
sin
2a
= 2 sin a
cos a |
and |
cos
2a
= cos2 a -
sin2 a |
|
|
|
 |
and |
 |
|
|
|
|
The double angle formula for the cosine function can be expressed
by sine or cosine function using the |
| identity
sin2 a
+ cos2 a
= 1, |
| |
cos
2a
=
2cos2 a -
1 |
or |
cos
2a
= 1
-
2sin2 a |
|
|
|
and |
1
+ cos
2a
=
2 cos2 a |
or |
1
-
cos
2a
=
2sin2 a |
|
|
|
| Trigonometric functions expressed by the half
angle |
| Substituting
a/2 in the double angle formulas we
obtain trigonometric functions expressed by the half angle, |
|
 |
and |
 |
|
|
|
 |
and |
 |
|
|
|
and |
|
 |
or |
 |
|
|
|
 |
or |
 |
|
|
|
| Trigonometric
identities
examples |
| Example:
Using known values, sin
60°
= Ö3/2
and sin 45°
= Ö2/2
evaluate sin 105°.
|
| 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° |
 |
|
| Example:
Express tan
3x in terms of tan
x. |
| Solution:
Using the sum formula and the double angle formula for the
tangent function, |
 |
|
| Example:
Prove the identity |
 |
|
|
|
|
| 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 |
 |
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Trigonometry
contents A |
|
|
 |
|
| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
