Answer:
The solution of the equation is Ф = 0 or Ф = 3π/2
Explanation:
* Lets revise some facts in trigonometry
- The identity sin² Ф + cos² Ф = 1
- By subtracting sin² Ф from both sides then cos² Ф = sin² Ф - 1
- In the rectangular plane the point (x , y) represents (cos Ф , sin Ф)
where x = cox Ф and y = sin Ф
- The point (1 , 0) lies on the positive part of x-axis means cos Ф = 1
and sin Ф = 0, then Ф = 0 or 2π
- The point (-1 , 0) lies on the negative part of x-axis means cos Ф = -1
and sin Ф = 0, then Ф = π
- The point (0 , 1) lies on the positive part of y-axis means cos Ф = 0
and sin Ф = 1, then Ф = π/2
- The point (0 , -1) lies on the negative part of y-axis means cos Ф = 0
and sin Ф = -1, then Ф = 3π/2
* Lets solve the problem
∵ sin Ф + 1 = cos² Ф
- To solve we must change cos² Ф to sin² Ф
∵ cos² Ф = sin² Ф - 1
- substitute cos² Ф in the equation by 1 - sin² Ф
∴ sin Ф + 1 = 1 - sin² Ф ⇒ add sin² Ф to both sides
∴ sin² Ф + sin Ф + 1 = 1 ⇒ subtract 1 from both sides
∴ sin² Ф + sin Ф = 0
- Take sin Ф as a common factor from both terms
∴ sin Ф (sin Ф + 1) = 0
- Equate each factor by 0
∴ sin Ф = 0 OR sin Ф + 1 = 0
- Remember 0 ≤ Ф < π
∵ sin Ф = 0 ⇒ from the information above
∴ Ф = 0
∵ sin Ф + 1 = 0 ⇒ subtract 1 from both sides
∴ sin Ф = -1
- From the information above
∴ Ф = 3π/2
* The solution of the equation is Ф = 0 or Ф = 3π/2