Tiêu đề 02. INVERSE TRIGONOMETRY FUCTION.pdf ✅

(c) If f(x) and g(x) are inverse of each other then fog(x) = x and gof(x) = x (d) If f and g are two bijections f:A → B, g : B → C, then the inverse of gof exists and (gof)−1 = f−1 o g−1 . (e) If f(x) and g(x) are inverse function of each other, then f(g(x)) = 1 g (x)  Formulas The basic inverse trigonometric formulas are as follows: Inverse Trig Functions Formulas Arcsine sin-1 (-x) = -sin-1 (x), x ∈ [-1, 1] Arccosine cos-1 (-x) = π -cos-1 (x), x ∈ [-1, 1] Arctangent tan-1 (-x) = -tan-1 (x), x ∈ R Arccotangent cot-1 (-x) = π – cot-1 (x), x ∈ R Arcsecant sec-1 (-x) = π -sec-1 (x), |x| ≥ 1 Arccosecant cosec-1 (-x) = -cosec-1 (x), |x| ≥ 1 INVERSE TRIGONOMETRY FUNCTIONS : Six inverse trigonometric functions are sin–1x, cos–1x, tan–1x, cosec–1x, sec–1x and cot–1x which are described in detail as below Arcsine Function Arcsine function is an inverse of the sine function denoted by sin-1x. It is represented in the graph as shown below: Domain -1 ≤ x ≤ 1 Range -π/2 ≤ y ≤ π/2 The branch with range [ −π 2 , π 2 ] is called the principal value branch. SOME OBSERVATIONS: (i) Sin and sin −1 are increasing functions on [ −π 2 , π 2 ] and [−1 ,1] respectively. ∴ θ1 < θ2 ⇒ sin θ1 < sin θ2 for all θ1 , θ2 ∈ [ −π 2 , π 2 ] And , x1 < x2 ⇒ sin−1 x1 < sin−1 x2 , for all x1 , x2 ∈ [ -1 , 1] (ii) The minimum and the maximum values of sin−1 x are −π 2 and π 2 respectively. (iii) sin−1 x attains the minimum value −π 2 at x = -1 and the maximum value π 2 at x = 1 Arccosine Function Arccosine function is the inverse of the cosine function denoted by cos-1x. It is represented in the graph as shown below: Therefore, the inverse of cos function can be expressed as; y = cos-1x (arccosine x) Domain & Range of arcsine function: Domain -1≤x≤1 Range 0 ≤ y ≤ π f(x) = 1/x Example sin−1 x is not equal to (sin x) −1 or 1 sin x Find the principal values of sin−1 ቀ −1 2 ቁ Solution: sin−1 ቀ −1 2 ቁ = − π 6 Example Note
The branch of cos−1 ∶ [ −1 , 1] → [0, π] is called the principal value branch and the value of cos−1 x lying in [ 0 , π ] for a given value of x ∈ [-1,1] is called the principal value. SOME OBSERVATIONS (i) The domain and range of cos−1 x are [ -1 ,1] and [ 0 , π ] respectively. (ii) Both cos and cos−1 are decreasing functions in their respective domains. ∴ θ1 < θ2 ⇒ cos θ1 > cos θ2 for all θ1 , θ2 ∈ [-1 ,1] (iii) The minimum and maximum value of cos−1 x are 0 and π respectively which are attained at 1 and -1 respectively i.e., cos−1(1) = 0 and cos−1(−1) = π Arctangent (Arctan) Function Arctangent function is the inverse of the tangent function denoted by tan-1x. It is represented in the graph as shown below: Therefore, the inverse of tangent function can be expressed as; y = tan-1x (arctangent x) Domain & Range of Arctangent: Domain -∞ < x < ∞ Range -π/2 < y < π/2 The branch with range ቀ −π 2 , π 2 ቁ is called the principal value branch of the function tan-1 SOME USEFUL OBSERVATIONS It is evident from the graphs of tan : ቀ −π 2 , π 2 ቁ → R and tan−1 : R → ቀ −π 2 , π 2 ቁ i.e., the curves y = tan x and y = tan−1 x that (i) −π 2 < tan−1 x < π 2 for all x ∈ R i.e., −π 2 and π 2 are minimum and maximum values of tan−1 x but it not does not attain these values. (ii) Both tan and tan−1 are increasing functions in their respective domains. ∴ θ1 < θ2 ⇒ tan θ1 < tanθ2 for all θ1 , θ2 ∈ ቀ −π 2 , π 2 ቁ And , x1 < x2 ⇒ tan−1 x1 < tan−1 x2 for all x1 , x2 ∈ R. Arccotangent (Arccot) Function Arccotangent function is the inverse of the cotangent function denoted by cot-1x. It is represented in the graph as shown below: Therefore, the inverse of cotangent function can be expressed as; y = cot-1x (arccotangent x) Domain & Range of Arccotangent: Domain -∞ < x < ∞ Range 0 < y < π The branch with range (0, π)is called the principal value branch of the function cot-1 SOME USEFUL OBSERVATION (i) Cot x is a decreasing function on ( 0 , π ). i.e., θ1 < θ2 ⇒ cot θ1 > cot θ2 for all θ1 , θ2 ∈ (0, π) (ii) cot−1 x is a decreasing function on R. i.e., x1 < x2 ⇒ cot−1 x1 > cot−1 x2 for all x1 , x2 ∈ R. (iii) For all x ∈ R ,the values of cot−1 lies between 0 and π (iv) cot−1 x does not attain its minimum value zero and maximum value π at points in R. ARCSECANT FUNCTION What is arcsecant (arcsec)function? Arcsecant function is the inverse of the secant function denoted by sec-1 x. It is represented in the graph as shown below: Find the principal values of cos−1 ቀ ξ3 2 ቁ Solution: cos−1 ቀ ξ3 2 ቁ = π 6 Example Find the principal values of tan−1൫−ξ3൯ Solution: tan−1൫−ξ3൯ = −π 3 Example