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The inverse trigonometric functions
 

The inverse trigonometric functions

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all abou the inverse trigonometri, derivative and integral of it

all abou the inverse trigonometri, derivative and integral of it

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    The inverse trigonometric functions The inverse trigonometric functions Document Transcript

    • THE INVERSE TRIGONOMETRIC FUNCTIONS By Group I AlfiramitaHertanti (1111040151) AmiraAzzahraYunus (1111040153) DEPARTEMEN OF MATHEMATIC EDUCATION FACULTY OF SCIENCE AND MATHEMATIC STATE UNIVERSITY OF MAKASSAR
    • A. DEFINITION In mathematics, the inverse trigonometric functions (occasionally called cyclometric function) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. Example The sign tan-1-1.374is employed to signify the angle whose tangent is -1.3674. And in General Sin-1 x means the angle whose sine is x Cos-1 x means the angle whose cosine is x Three pointsshould be noted. 1) Sin-1 x stand for an angle: thus sin-1 ½ = 30o 2) The “-1” is not an index, but merely a sign to denote inversenotation. 3) (sin x)-1 is not used, because it mean the reciprocal of sin x and this is cosec x. If a functionfis one-to-one on its domain, then f has aninverse function, denoted by f−1, such that y=f(x) if and onlyif f−1 (y)= x. The domain of f−1 is the range of f. The basicidea is that f-1“undoes” what f does, andvice versa. In otherwords, f−1 (f(x)) =x for all xin the domain of f, and f (f−1(y)) =y for all yin the range of f. They are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used inengineering, navigation, physics, and geometry.
    • B. GRAPHICS a. Inverse Sine and Inverse Cosine To define the inverse functions for sine and cosine(sometimes called the arcsineand arccosine and denoted byy=arcsinxor y = arccosine x), the domains of these functions are restricted. The restriction thatisplaced on the domain values of the cosine function is (see Figure 7-2). This restricted function is called Cosine. Note the capital “C” in Cosine The inverse cosine function is defined as the inverse of the restriced Cosine Function . Therefore, Identities for the cosine and inverse cosine: The inverse sine function’s development is similar to that cosine. The restriction that is placed on the domain values of the sine function is This restricted functioniscalled Sine(seeFigure 7-4). Note the capital "S" in Sine.
    • The inversesine function(seeFigure 7-5) is defined as the inverse of the restrictedSine functiony = Sin x, Therefore, and Identities for the sineand inverse sine:
    • Thegraphs of the functions y= Cos x and y = Cos-1 x are reflections of each otherabout the liney = x. The graphs of the functions y = Sin x andy = Sin-1x are also reflections of each other about the liney = x (see Figure 7-6). EXAMPLE 1 : Using Figure 7-7, find the exact value of Thus, . Example 2 : Using Figure 7-8, Find the exact value of Thus,
    • Other Inverse Trigonometric Functions To define theinversetangent, the domain ofthe tangent must be restricted to This restricted function is Called Tangent (See Figure 7-9). Note the capital “T” in Tangent. The inverse tangent function (see Figure7-10) is defined as theinverseofthe restricted Tangent function y = Tan x, Therefore,
    • Identitiesforthe tangent and inverse tangent: The inverse tangent, inverse secant, and inverse cosecantfunctions are derived from the restricted Sine, Cosine, and Tangent functions. The graphs of these functionsare shown in Figure 7-11.
    • Trigonometriidentities involving inverse cotangent, inverse secant, and inverse cosecant:
    • EXAMPLE 2. Calculate (a) (b) (c) (d) SOLUTION (a) (b) (c) (d) EXAMPLE 3. Calculate (a) (b) (c) (d) SOLUTION (a) (b) (c) (d)
    • C. PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTION Four Useful Identities Theorem A gives some useful identities. You can recall them by reference to the triangles in Figure 7. Theorem A (i) (ii) (iii) (iv) EXAMPLE 4 Calculate SOLUTION Recall the double-angel identity
    • From the inverse Function Theorem (Theorem 6.2B), we conclude that sin-1, cos-1, tan-1, cot-1, csc-1and sec-1 are differentiable. Our aim is to find formulas for their derivatives. Theorem B. Derivatives of Inverse Trigonometric Function (i) (ii) -1 < x < 1 -1 < x < 1 (iii) (iv) (v) (vi) EXAMPLE 5Find SOLUTION We use Theorem B(i) and the Chain Rule. Every differential formula leads to an integration formula, a matter we wiil say much more about in the next chapter. In Particular, 1. 2. 3. These integration formulas can be generalized slightly to the following: 1. 2. 3.
    • EXAMPLE 6. Evaluate SOLUTION Think Of . Then +C Expression as definite Integral Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral :
    • When x equals 1, the integral with limited domains are improper integrals, but still well-defined. EXAMPLE 7. Evaluate SOLUTION D. SUMMERY The Inverse Trigonometric Function are the inverse function of trigonometric function with suitably restricted domains. They are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. The inverse cosine function is defined as the inverse of the restriced Cosine Function . Therefore, Theinverse sine function (see Figure 7-5) isdefined as the inverse of the restricted Sine functiony = Sin x, Identities for the tangent and inverse tangent: Trigonometric identities involving inverse cotangent, inverse secant, and inverse cosecant:
    • From the inverse Function Theorem, we conclude that sin-1, cos-1, tan-1, cot-1, csc-1and sec-1 are differentiable.
    • REFERENCE Kay, David. 2001. CliffsQuickReviewTM Trigonometry. Hungry Minds, Inc : New York. Corral, Michael. 2009.Trigonometry.GNU Free Documentation License Purcell, Edwind J. 2007. Calculus Ninth Edition. Pearson Education,Ltd : London. http://en.wikipedia.org/wiki/inverse_trigonometric_functions