| Solve the equation | ||
| Arccos x - Arcsin x | = |
5π
6
|
 |
- the rule "Must have" of Mr.Zhang ® | |
|
| Solution | |||||
| From the problem, take cos both side; | |||||
| Arccos x | = | Arcsin x | + |
5π
6
| |
| cos(Arccos x) | = | cos(Arcsin x | + |
5π
6
|
) |
| x | = | cos(Arcsin x | + |
5π
6
|
) |
From identity;
| x | = | cos ( |
5π
6
|
+ | Arcsin x ) | ||||
| x | = | cos |
5π
6
|
cosθ | - sin |
5π
6
|
sinθ | ||
(∴ Arcsin x = θ)
|
|||||||
| θ=Arcsinx sinθ=sin(Arcsinx) |
|
cos(Arcsinx) = cos θ
=
| x | = |
-√3
2
|
( √1-x2) | - ½ x |
| 2x | = | -√3 ( √1-x2) | - x |
| 3x | = | -√3 ( √1-x2) |
| 9x2 | = | 3 ( 1-x2) |
| 3x2 | = | ( 1-x2) |
| 4x2 | = | 1 |
| x2 | = | ¼ |
| x | = | (+/-)½ |
Verify x = ½
| Arccos½=Arcsin½+ |
5π
6
|
| 60o=30o+150o | |
| 60o=180o | →false |
Verify x = (-)½
| Arccos(-)½=Arcsin(-)½+ |
5π
6
|
| 120o=-30o+150o | |
| 120o=120o | →right |
| Ans. x = - ½ |
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