VIEWS: 19 PAGES: 12 CATEGORY: Nutrition & Healthy Eating POSTED ON: 5/30/2010 Public Domain
L26 The Inverse Trigonometric Functions The six trigonometric functions are not one-to-one on their domains. To define the inverse of a trigonometric function, we consider the function on the restricted domain where it is one-to-one. The restricted domain we will call the Interval of Definition. Intervals of Definition and Inverses of the Trigonometric Functions: f ( x ) = sin x Interval of Definition: f −1 ( x ) = sin −1 x : Domain: Range: Note: f −1 ( x ) = sin −1 x is an odd function. 312 f ( x ) = cos x Interval of Definition: f −1 ( x ) = cos −1 x : Domain: Range: f ( x ) = tan x Interval of Definition: f −1 ( x ) = tan −1 x : Domain: Range: Note: f −1 ( x ) = tan −1 x is an odd function. 313 f ( x ) = cot x Interval of Definition: f −1 ( x ) = cot −1 x : Domain: Range: Notes: The domains of the inverse trigonometric functions are the ranges of the corresponding trigonometric function. The ranges of the inverse trigonometric functions are the Intervals of Definitions of the corresponding trigonometric functions. Important: Values of an inverse trigonometric function are the angles from the Interval of Definition (see the ranges of the inverse functions). 314 Cancellation Rules (Part I) Identities: sin ( sin −1 x ) = x −1 ≤ x ≤ 1 cos ( cos −1 x ) = x −1 ≤ x ≤ 1 tan ( tan −1 x ) = x −∞ < x < +∞ cot ( cot −1 x ) = x −∞ < x < +∞ Example: Evaluate: sin ( sin −1 (1/ 4) ) ⎛ 3⎞ cos ⎜ cos −1 ⎟ ⎝ 2⎠ Definitions of the inverse trigonometric functions come from the Cancellation Rules (Part I): The inverse sine, cosine, tangent or cotangent of a real number x is the angle from the Interval of Definition whose sine, cosine, tangent or cotangent is equal to x. Note: Number x must be in the domain of the inverse function. 315 Sometimes, one can easily find the exact value of an inverse function by using the definitions above. In some complicated cases, however, it is useful to know the extended procedure of Finding Exact Value of the Inverse Trigonometric Function Example: Find sin −1 x ( −1 ≤ x ≤ 1). 1. Denote: sin −1 x = θ 2. Compose sine function to both sides to cancel the inverse: x = sin θ 3. Pick up the appropriate value for θ from the Interval of Definition. 316 Example: Find the exact values (without a calculator) sin −1 1 = ⎛ 1⎞ cos −1 ⎜ − ⎟ = ⎝ 2⎠ ⎛ 3⎞ arcsin ⎜ − ⎟ = cos −1 1 = ⎝ 2 ⎠ tan −1 (−1) = ( cot −1 − 3 = ) π π π π π π π π x − − − − 0 2 3 4 6 6 4 3 2 3 2 1 1 2 3 sin x −1 − − − 0 1 2 2 2 2 2 2 1 1 tan x ---- − 3 −1 − 0 1 3 ---- 3 3 π π π π 2π 3π 5π x 0 6 4 3 2 3 4 6 π 3 2 1 1 2 3 cos x 1 0 − − − −1 2 2 2 2 2 2 1 1 cot x ---- 3 1 0 − −1 − 3 ---- 3 3 317 The Inverse Secant Function f ( x) = sec x Interval of Definition: f −1 ( x) = sec−1 x Domain: Range: HA: 318 The Inverse Cosecant Function f ( x) = csc x Interval of Definition: f −1 ( x) = csc−1 x Domain: Range: HA: 319 Note: The functions sec x and csc x have the same Intervals of Definition as cos x and sin x , respectively, except for the points where secant and cosecant are undefined. Example: Find the exact values without a calculator. csc −1 2 = sec−1 1 = ⎛ 2 ⎞ csc −1 ⎜ − ⎟ ⎝ 3⎠ sec −1 ( −2 ) 320 Cancellation Rules (Part II) Identities: (Interval of Definition) ⎡ π π⎤ sin −1 ( sin θ ) = θ θ ∈ ⎢− , ⎥ ⎣ 2 2⎦ ⎛ π π⎞ tan −1 ( tan θ ) = θ θ ∈⎜ − , ⎟ ⎝ 2 2⎠ cos −1 ( cosθ ) = θ θ ∈ [ 0, π ] cot −1 ( cot θ ) = θ θ ∈ ( 0, π ) Note: If θ is not in the Interval of Definition, the Identities do not hold. In order to extend the use of the cancellation rules for any θ in the Domain of the trigonometric function, we must replace angle θ with the unique angle θ1 from the Interval of Definition such that sin θ1 = sin θ . The angle θ1 can be found by using Periodic Properties, Even-Odd Properties, graphs and/or the Unit Circle. The modified Cancellation Rule is sin −1 (sin θ ) = sin −1 (sin θ1 ) = θ1 321 Example: Evaluate: ⎛ π⎞ cos −1 ⎜ cos ⎟ ⎝ 5⎠ sin −1 ( sin(14π / 5) ) cos −1 ( cos(6π / 5) ) ⎛ 27π ⎞ tan −1 ⎜ tan ⎟ ⎝ 4 ⎠ 322 Example: Give each value without using a calculator. ⎛ ⎛ 1 ⎞⎞ sin ⎜ cos −1 ⎜ − ⎟ ⎟ ⎝ ⎝ 4 ⎠⎠ Example: Write as an expression in u. sin ( tan −1 u ) = cos ( tan −1 u ) = 323