Question

log of sin X to the base (1/√2) is greater than 0 and X belongs to a closed interval of [0,4π]. Find the number of values of X which are integral multiples of π/4​

Answer 1

Answer:

Note: Principal value range of all Inverse Circular function is very important as the function is defined only in this range.

Pause: Odd function are defined as f(–x) = –f(x) and even function as f(–x) = f(x).

The inverse circular functions are defined as below:-

1. sin–1 (–x) = –sin–1 x, –1 < x < 1 Odd function

2. cos–1 (–x) = π –cos–1 x, –1 < x < 1 Neither odd nor even

3. tan–11 (–x) = –tan–1 x, x ∈ R Odd function

4. cot–1 (–x) = π – cot–1 x, x ∈ R Neither odd nor even

5. cosec–1 (–x) = –cosec–1 x, x < –1 or x > 1 Odd function

6. sec–1 (–x) = π –sec–1 x, x < –1 or x > 1 Neither odd nor even

Let us see the proof of any one of the above.

Proof 2:

Let cos-1 (–x) = θ, then cos θ = –x

or, – cos θ = x or cos (π – θ) = x

or, π – θ = cos-1 x or cos-1 (–x) = π –cos-1 x

Similarly we can prove other results.

Caution: Instead of taking cos (π – θ) equal to – cos θ, we could have taken cos (π + θ). We opt for cos (π – θ) because (π – θ) lies in a principal value range i.e. 0 ≤ cos-1 x ≤ π.