|
Trigonometry |
|
|
|
Trigonometric functions expressed by the
tangent of the half angle |
Half angle formulas |
Trigonometric functions expressed by the
cosine of the double angle |
Trigonometric identities,
examples |
|
|
|
|
|
Trigonometric functions expressed by the
tangent of the half angle |
Replacing a
by a/2
in the above identities, we get
|
|
|
Half angle formulas |
Using
the identities in which trigonometric functions are expressed by
the half angle, |
|
and
applying the definitions of the functions, tangent and cotangent |
|
|
Trigonometric functions expressed by the
cosine of the double angle
|
Replacing a/2
by a
in the above identities, we get
|
|
|
Trigonometric identities,
examples |
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. |
|
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) |
|
Example:
Prove the identity |
![](TrigEx19.gif) |
|
|
|
Example:
Prove the identity |
![](TrigEx20.gif) |
|
|
|
Example:
If tan
a
= 3/4, find tan
a/2.
|
Solution:
Use formula |
![](TrigEx21.gif) |
to express tan
a/2
in terms of tan
a. |
|
![](TrigEx21A.gif) |
|
Example:
Prove the identity |
![](TrigEx22.gif) |
|
Solution:
Substitute |
![](TrigEx22A.gif) |
then |
|
![](TrigEx22B.gif) |
|
Example:
Prove the identity |
![](TrigEx25.gif) |
|
Solution:
Using the formula for the sum of the tangent |
![](TrigEx25A.gif) |
|
![](TrigEx25B.gif) |
|
Example:
Prove that |
![](TrigEx26.gif) |
|
Solution: Replace
sin a
by cos (p/2
-
a)
and cos a
by sin (p/2
-
a)
and use the sum to product
formula
|
![](TrigEx26A.gif) |
|
|
|
|
|
|
|
|
Pre-calculus contents
F |
|
![](HOME1.gif) |
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |