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General and Principal Values of sin1 x

What are the general and principal Values of sin1 x?

What is sin1 ½?

We know that sin (30°) = ½.

⇒ sin1 (1/2) = 30° or π6.

Again, sin θ = sin (π - π6)

⇒ sin θ = sin (5π6)

⇒ θ = 5π6or 150°

Again, sin θ = 1/2

⇒ sin θ = sin π6

⇒ sin θ = sin (2π + π6)

⇒ sin θ = sin (13π6)

⇒ θ = 13π6 or 390°

Therefore, sin (30°) = sin (150°) = sin (390°) and so on, and, sin (30°) = sin (150°) = sin (390°) = ½.

In other ward we can say that,

sin (30° + 360° n) = sin (150° + 360° n) = ½, where, where n = 0, ± 1, ± 2, ± 3, …….

And in general, if sin θ = ½ = sin π6 then θ = nπ + (- 1)n π6, where n = 0 or any integer.

Therefore, if sin θ = 1/2 then θ = sin1 ½ = π6 or 5π6 or 13π6

Therefore in general, sin1  (½) = θ = nπ + (-1) n π6 and the angle nπ + (- 1)n π6 is called the general value of sin1 ½.

The positive or negative least numerical value of the angle is called the principal value

In this case the π6 is the least positive angle. Therefore, the principal value of sin1 ½ is π6.

Let sin θ = x and - 1 ≤ x ≤ 1

x ⇒ sin {nπ + (- 1)n θ}, where n = 0, ± 1, ± 2, ± 3, …….

Therefore, sin1 x = nπ + (- 1)n θ, where n = 0, ± 1, ± 2, ± 3, …….

For the above equation we can say that sin1 x may have infinitely many values.

Let – π2 ≤ α ≤ π2, where α is positive or negative smallest numerical value and satisfies the equation sin θ = x then the angle α is called the principal value of sin1 x.

Therefore, the general value of sin1 x is nπ + (- 1)n θ, where n = 0, ± 1, ± 2, ± 3, …….

The principal value of sin1 x is α, where - π2 ≤ α ≤ π2 and α satisfies the equation sin θ = x.

For example, principal value of sin1 (-32) is -π3and its general value is nπ + (- 1)n ∙ (-π3) = nπ - (- 1)nπ3

Similarly, principal value of sin1 (32) is (π3) and its general value is nπ + (- 1)n (π3) = nπ - (- 1)nπ6

 Inverse Trigonometric Functions







11 and 12 Grade Math

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