Any trigonometric equation of the form cos-a - a- -1- 1- will have principal solutions-

Any trigonometric equation cosθ=a ∀ a∈ [ 1, 1]. Case 1: a≥ 0⇒ 0≤cosθ≤ 1 Case 2: a lt; 0⇒ 1≤cosθ lt;0 Now, as we can see from both the cases, cosθ=a ∀ a∈ ...

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Standard XII

Mathematics

General Solution of cos theta = cos alpha

Any trigonome...

Question

Any trigonometric equation of the form $cos θ = a \forall a \in (- 1, 1),$ will have principal solution(s).

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Solution

The correct option is C $2$
Any trigonometric equation $cos θ = a \forall a \in [- 1, 1] .$
Case 1: $a \geq 0 \Rightarrow 0 \leq cos θ \leq 1$

Case 2: $a < 0 \Rightarrow - 1 \leq cos θ < 0$

Now, as we can see from both the cases, $cos θ = a \forall a \in [- 1, 1]$ will have two solution for $θ \in [0, 2 π)$
Thus, there will be $2$ principal solutions.

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Any trigonometric equation of the form cosθ=a ∀ a∈(−1,1), will have principal solution(s).

The correct option is C 2Any trigonometric equation cosθ=a ∀ a∈[−1,1]. Case 1: a≥0⇒0≤cosθ≤1 Case 2: a<0⇒−1≤cosθ<0 Now, as we can see from both the cases, cosθ=a ∀ a∈[−1,1] will have two solution for θ∈[0,2π) Thus, there will be 2 principal solutions.

Any trigonometric equation of the form $cos θ = a \forall a \in (- 1, 1),$ will have principal solution(s).