Question

explain the need and importance of Preamble in preparing map​

Answer 1

Answer:

Prove that the general solution of sin θ = sin ∝ is given by θ = nπ + (-1)[Math Processing Error] ∝, n ∈ Z.

Solution:

We have,

sin θ = sin ∝

⇒ sin θ - sin ∝ = 0

⇒ 2 cos [Math Processing Error] sin [Math Processing Error] = 0

Therefore either cos [Math Processing Error] = 0 or, sin [Math Processing Error] = 0

Now, from cos [Math Processing Error] = 0 we get, [Math Processing Error] = (2m + 1)[Math Processing Error], m ∈ Z

⇒ θ = (2m + 1)π - ∝, m ∈ Z i.e., (any odd multiple of π) - ∝ ……………….(i)

And from sin [Math Processing Error] = 0 we get,

[Math Processing Error] = mπ, m ∈ Z

⇒ θ = 2mπ + ∝, m ∈ Z i.e., (any even multiple of π) + ∝ …………………….(ii)

Now combining the solutions (i) and (ii) we get,

θ = nπ + (-1)[Math Processing Error] ∝, where n ∈ Z.

Hence, the general solution of sin θ = sin ∝ is θ = nπ + (-1)[Math Processing Error] ∝, where n ∈ Z.

Note: The equation csc θ = csc ∝ is equivalent to sin θ = sin ∝ (since, csc θ = [Math Processing Error] and csc ∝ = [Math Processing Error]). Thus, csc θ = csc ∝ and sin θ = sin ∝ have the same general solution.

Hence, the general solution of csc θ = csc ∝ is θ = nπ + (-1)[Math Processing Error] ∝, where n ∈ Z.