inverse trigonometry formula derivation

Trigonometric Identity 3. Detailed step by step solutions to your Derivatives of inverse trigonometric functions problems online with our math solver and calculator. In order to derive the derivatives of inverse trig functions we’ll need the formula from the last section relating the derivatives of inverse functions. For example, the sine function x = φ(y) = siny is the inverse function for y = f (x) = arcsinx. 2 The graph of y = sin x does not pass the horizontal line test, so it has no inverse. And similarly for each of the inverse trigonometric functions. Derivatives of inverse trigonometric functions Calculator online with solution and steps. Find the derivative of f given by f (x) = sec–1 assuming it exists. The derivatives of $$6$$ inverse trigonometric functions considered above are consolidated in the following table: In the examples below, find the derivative of the given function. Use the formula given above to nd the derivative of f1. Differentiate functions that contain the inverse trigonometric functions arcsin(x), arccos(x), and arctan(x). ddx(sin−1x)=11–x2{ \frac{d}{dx}(sin^{-1}x) = \frac{1}{\sqrt{1 – x^2}}} dxd​(sin−1x)=1–x2​1​ Also, ddx(cos−1x)=−11–x2{ \frac{d}{dx}(cos^{-1}x) = \frac{-1}{\sqrt{1 – x^2}}} dxd​(cos−1x)=1–x2​−1​ ddx(tan−1x)=11+x2{ \frac{d}{dx}(tan^{-1}x) = \frac{1}{1 + x^2}} dxd​(tan−1x)=1+x21​ ddx(cosec−1x)=−1mod(x).x2–1{ \frac{d}{dx}(cosec^{-1}x) = \frac{-1}{mod(x).\sqrt{x… $$\frac{d}{dx}(\textrm{arccot } x) = \frac{-1}{1+x^2}$$, Finding the Derivative of the Inverse Secant Function, $\displaystyle{\frac{d}{dx} (\textrm{arcsec } x)}$. The derivative of y = arccos x. For example, the domain for $$\arcsin x$$ is from $$-1$$ to $$1.$$ The range, or output for $$\arcsin x$$ is all angles from $$– \large{\frac{\pi }{2}}\normalsize$$ to $$\large{\frac{\pi }{2}}\normalsize$$ radians. }\], ${y’\left( x \right) }={ {\left( {\arctan \frac{{x + 1}}{{x – 1}}} \right)^\prime } }= {\frac{1}{{1 + {{\left( {\frac{{x + 1}}{{x – 1}}} \right)}^2}}} \cdot {\left( {\frac{{x + 1}}{{x – 1}}} \right)^\prime } }= {\frac{{1 \cdot \left( {x – 1} \right) – \left( {x + 1} \right) \cdot 1}}{{{{\left( {x – 1} \right)}^2} + {{\left( {x + 1} \right)}^2}}} }= {\frac{{\cancel{\color{blue}{x}} – \color{red}{1} – \cancel{\color{blue}{x}} – \color{red}{1}}}{{\color{maroon}{x^2} – \cancel{\color{green}{2x}} + \color{DarkViolet}{1} + \color{maroon}{x^2} + \cancel{\color{green}{2x}} + \color{DarkViolet}{1}}} }= {\frac{{ – \color{red}{2}}}{{\color{maroon}{2{x^2}} + \color{DarkViolet}{2}}} }= { – \frac{1}{{1 + {x^2}}}. To determine the sides of a triangle when the remaining side lengths are known. \[{y^\prime = \left( {\arctan \frac{1}{x}} \right)^\prime }={ \frac{1}{{1 + {{\left( {\frac{1}{x}} \right)}^2}}} \cdot \left( {\frac{1}{x}} \right)^\prime }={ \frac{1}{{1 + \frac{1}{{{x^2}}}}} \cdot \left( { – \frac{1}{{{x^2}}}} \right) }={ – \frac{{{x^2}}}{{\left( {{x^2} + 1} \right){x^2}}} }={ – \frac{1}{{1 + {x^2}}}. Thus, an equation of the tangent line is . }$, $\require{cancel}{y^\prime = \left( {\arcsin \left( {x – 1} \right)} \right)^\prime }={ \frac{1}{{\sqrt {1 – {{\left( {x – 1} \right)}^2}} }} }={ \frac{1}{{\sqrt {1 – \left( {{x^2} – 2x + 1} \right)} }} }={ \frac{1}{{\sqrt {\cancel{1} – {x^2} + 2x – \cancel{1}} }} }={ \frac{1}{{\sqrt {2x – {x^2}} }}. Purely algebraic derivations are longer. }$, ${y^\prime = \left( {\text{arccot}\,{x^2}} \right)^\prime }={ – \frac{1}{{1 + {{\left( {{x^2}} \right)}^2}}} \cdot \left( {{x^2}} \right)^\prime }={ – \frac{{2x}}{{1 + {x^4}}}. By deﬁnition of an inverse function, we want a function that satisﬁes the condition x =sinhy = e y−e− 2 by deﬁnition of sinhy = ey −e− y 2 e ey = e2y −1 2ey. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We'll assume you're ok with this, but you can opt-out if you wish. In this section we are going to look at the derivatives of the inverse trig functions. Examples of implicit functions: ln(y) = x2; x3 +y2 = 5, 6xy = 6x+2y2, etc. Free tutorial and lessons. That's why I think it's worth your time to learn how to deduce them by yourself. You also have the option to opt-out of these cookies. Solution We have f0(x) = 2x, so that f0(f1(x)) = 2 p x. The concepts of inverse trigonometric functions is also used in science and engineering. Click or tap a problem to see the solution. Similarly, inverse functions of the basic trigonometric functions are said to be inverse trigonometric functions. Integrals Involving the Inverse Trig Functions. Consider, the function y = f (x), and x = g (y) then the inverse function is written as g = f -1, This means that if y=f (x), then x = f -1 (y). The process for finding the derivative of \arccos x is almost identical to that used for \arcsin x: Suppose \arccos x = \theta. The inverse of g is denoted by ‘g -1’. INVERSE TRIGONOMETRIC FUNCTIONS OBJECTIVES: derive the formula for the derivatives of the inverse trigonometric functions; apply the derivative formulas to solve for the derivatives of inverse trigonometric functions; and solve problems involving derivatives of inverse trigonometric functions Differentiation of inverse trigonometric functions is a small and specialized topic. Derivatives of Inverse Trigonometric Functions The derivatives of the inverse trigonometric functions can be obtained using the inverse function theorem. In trigonometry class 12, we study trigonometry which finds its application in the field of astronomy, engineering, architectural design, and physics.Trigonometry Formulas for class 12 contains all the essential trigonometric identities which can fetch some direct questions in competitive exams on the basis of formulae. }$, ${y^\prime = \left( {\text{arccot}\frac{1}{{{x^2}}}} \right)^\prime }={ – \frac{1}{{1 + {{\left( {\frac{1}{{{x^2}}}} \right)}^2}}} \cdot \left( {\frac{1}{{{x^2}}}} \right)^\prime }={ – \frac{1}{{1 + \frac{1}{{{x^4}}}}} \cdot \left( { – 2{x^{ – 3}}} \right) }={ \frac{{2{x^4}}}{{\left( {{x^4} + 1} \right){x^3}}} }={ \frac{{2x}}{{1 + {x^4}}}.}$. -1/ (1+ x2 ) arcsecx = sec-1x. Trigonometric functions of inverse trigonometric functions are tabulated below. Table Of Derivatives Of Inverse Trigonometric Functions. These functions are widely used in fields like physics, mathematics, engineering, and other research fields. Hence, it is essential to learn the derivative formulas for evaluating the derivative of every inverse trigonometric function. Similar to the method described for sin-1x, one can calculate all the derivative of inverse trigonometric functions. These cookies do not store any personal information. Differentiating implicitly, I get … Implicitly differentiating with respect to $x$ yields Lessons On Trigonometry Inverse trigonometry Trigonometric Derivatives Calculus: Derivatives Calculus Lessons. The beauty of this formula is that we don’t need to actually determine () to find the value of the derivative at a point. The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. In the same way for trigonometric functions, it’s the inverse trigonometric functions. Thus, DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS 2. Thus, Finally, plugging this into our formula for the derivative of $\arccos x$, we find, Finding the Derivative of the Inverse Tangent Function, $\displaystyle{\frac{d}{dx} (\arctan x)}$. The Before reading this, make sure you are familiar with inverse trigonometric functions. So let's apply the derivative operator, d/dx on the left-hand side, d/dx on the right-hand side. Same idea for all other inverse trig functions Implicit Diﬀerentiation Use whenever you need to take the derivative of a function that is implicitly deﬁned (not solved for y). Indefinite integrals of inverse trigonometric functions. The inverse functions exist when appropriate restrictions are placed on the domain of the original functions. Here, we suppose $\textrm{arcsec } x = \theta$, which means $sec \theta = x$. The derivative of y = arctan x. sin, cos, tan, cot, sec, cosec. Put = sin 1(x) and note that 2[ ˇ=2;ˇ=2]. These cookies will be stored in your browser only with your consent. The formulas may look complicated, but I think you will find that they are not too hard to use. Using this technique, we can find the derivatives of the other inverse trigonometric functions: ${{\left( {\arccos x} \right)^\prime } }={ \frac{1}{{{{\left( {\cos y} \right)}^\prime }}} }= {\frac{1}{{\left( { – \sin y} \right)}} }= {- \frac{1}{{\sqrt {1 – {{\cos }^2}y} }} }= {- \frac{1}{{\sqrt {1 – {{\cos }^2}\left( {\arccos x} \right)} }} }= {- \frac{1}{{\sqrt {1 – {x^2}} }}\;\;}\kern-0.3pt{\left( { – 1 < x < 1} \right),}\qquad$, ${{\left( {\arctan x} \right)^\prime } }={ \frac{1}{{{{\left( {\tan y} \right)}^\prime }}} }= {\frac{1}{{\frac{1}{{{{\cos }^2}y}}}} }= {\frac{1}{{1 + {{\tan }^2}y}} }= {\frac{1}{{1 + {{\tan }^2}\left( {\arctan x} \right)}} }= {\frac{1}{{1 + {x^2}}},}$, ${\left( {\text{arccot }x} \right)^\prime } = {\frac{1}{{{{\left( {\cot y} \right)}^\prime }}}}= \frac{1}{{\left( { – \frac{1}{{{\sin^2}y}}} \right)}}= – \frac{1}{{1 + {{\cot }^2}y}}= – \frac{1}{{1 + {{\cot }^2}\left( {\text{arccot }x} \right)}}= – \frac{1}{{1 + {x^2}}},$, ${{\left( {\text{arcsec }x} \right)^\prime } = {\frac{1}{{{{\left( {\sec y} \right)}^\prime }}} }}= {\frac{1}{{\tan y\sec y}} }= {\frac{1}{{\sec y\sqrt {{{\sec }^2}y – 1} }} }= {\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}.}$. Table Of Derivatives Of Inverse Trigonometric Functions. 1 - Derivative of y = arcsin (x) The differentiation formula for f -1 can not be applied to the inverse of the cubing function at 0 since we can not divide by zero. Like before, we differentiate this implicitly with respect to $x$ to find, Solving for $d\theta/dx$ in terms of $\theta$ we quickly get, This is where we need to be careful. The following inverse trigonometric identities give an angle in different ratios. To be a useful formula for the derivative of $\arccos x$ however, we would prefer that $\displaystyle{\frac{d\theta}{dx} = \frac{d}{dx} (\arccos x)}$ be expressed in terms of $x$, not $\theta$. Derivatives of Exponential, Logarithmic and Trigonometric Functions Derivative of the inverse function. They are arcsin x, arccos x, arctan x, arcsec x, and arccsc x. We prove the formula for the inverse sine integral. Then . Notice that f '(x)=3x 2 and so f '(0)=0. The derivative of arccos in trigonometry is an inverse function, and you can use numbers or symbols to find out the answer to a problem. The domains of the other trigonometric functions are restricted appropriately, so that they become one-to-one functions and their inverse can be determined. In mathematics, inverse usually means the opposite. Then it must be the case that. But opting out of some of these cookies may affect your browsing experience. the graph of f(x) passes the horizontal line test), then f(x) has the inverse function f 1(x):Recall that fand f 1 are related by the following formulas y= f 1(x) ()x= f(y): As such. This website uses cookies to improve your experience while you navigate through the website. However, some teachers use the power of -1 instead of arc to express them. In the last formula, the absolute value $$\left| x \right|$$ in the denominator appears due to the fact that the product $${\tan y\sec y}$$ should always be positive in the range of admissible values of $$y$$, where $$y \in \left( {0,{\large\frac{\pi }{2}\normalsize}} \right) \cup \left( {{\large\frac{\pi }{2}\normalsize},\pi } \right),$$ that is the derivative of the inverse secant is always positive. To be a useful formula for the derivative of $\arcsin x$ however, we would prefer that $\displaystyle{\frac{d\theta}{dx} = \frac{d}{dx} (\arcsin x)}$ be expressed in terms of $x$, not $\theta$. In both, the product of $\sec \theta \tan \theta$ must be positive. Therefore, cot–1= 1 x 2 – 1 = cot–1 (cot θ) = θ = sec–1 x, which is the simplest form. According to the inverse relations: y = arcsin x implies sin y = x. Inverse trigonometric functions formula Summary: In this article you are going to learn all the inverse trigonometric functions formula also known as Inverse Circular Function. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS. Of course $|\sec \theta| = |x|$, and we can use $\tan^2 \theta + 1 = \sec^2 \theta$ to establish $|\tan \theta| = \sqrt{x^2 - 1}$. TRANSCENDENTAL FUNCTIONS Kinds of transcendental functions: 1.logarithmic and exponential functions 2.trigonometric and inverse trigonometric functions 3.hyperbolic and inverse hyperbolic functions Note: Each pair of functions above is an inverse to each other. Such that f (g (y))=y and g (f (y))=x. 3 Definition notation EX 1 Evaluate these without a calculator. Mathematical articles, tutorial, examples. Watch Queue Queue. For example, arcsin x is the same as sin ⁡ − 1 x \sin^{-1} x sin − 1 x. •Limits of arctan can be used to derive the formula for the derivative (often an useful tool to understand and remember the derivative formulas) Derivatives of Inverse Trig Functions (��−1)= 1 1−�2 (���−1)=-1 1−�2 In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Definitions as infinite series. Derivative of Inverse Trigonometric functions The Inverse Trigonometric functions are also called as arcus functions, cyclometric functions or anti-trigonometric functions. Inverse Trigonometry Functions and Their Derivatives. Dividing both sides by $\sec^2 \theta$ immediately leads to a formula for the derivative. e2y −2xey −1=0. 3 Definition notation EX 1 Evaluate these without a calculator. Here, the list of derivatives of inverse trigonometric functions with proofs in differential calculus. Some people find using a drawing of a triangle helps them figure out the solutions easier than using equations. Derivative Formulas. This implies. Differntiation formulas of basic logarithmic and polynomial functions are also provided. The inverse trigonometric functions are also called arcus functions or anti trigonometric functions. Solved exercises of Derivatives of inverse trigonometric functions. Integrals Involving the Inverse Trig Functions. In the list of problems which follows, most problems are average and a few are somewhat challenging. The derivative of y = arccsc x. I T IS NOT NECESSARY to memorize the derivatives of this Lesson. Section 3-7 : Derivatives of Inverse Trig Functions For each of the following problems differentiate the given function. Integrals that Result in Inverse Trigonometric Functions. La fonction cotangente est la fonction définie par : ( remarque c'est l'inverse de la tangente ) elle est définie pour toute valeur de x qui n'annule pas sin x, elle n' est donc définie pour x = k πavec k . SOLUTION 10 : Determine the equation of the line tangent to the graph of at x = e. If x = e, then , so that the line passes through the point . (ey)2 −2x(ey)−1=0. The derivative of y = arcsin x. of a function). The derivatives of the above-mentioned inverse trigonometric functions follow from trigonometry identities, implicit differentiation, and the chain rule. When we integrate to get Inverse Trigonometric Functions back, we have use tricks to get the functions to look like one of the inverse trig forms and then usually use U-Substitution Integration to perform the integral.. Inverse Trigonometry. $$-csc^2 \theta \cdot \frac{d\theta}{dx} = 1$$ Well, on the left-hand side, we would apply the chain rule. Logarithmic forms. Detailed step by step solutions to your Derivatives of inverse trigonometric functions problems online with our math solver and calculator. Therefore, the identity is true for all such that, 0° < a ≤ 90°. Video transcript ... What I want to do is take the derivative of both sides of this equation right over here. Inverse trigonometric functions have various application in engineering, geometry, navigation etc. Let us begin this last section of the chapter with the three formulas. If we restrict the domain (to half a period), then we can talk about an inverse function. Another method to find the derivative of inverse functions is also included and may be used. For all such that, Implicitly differentiating the above with respect to $x$ running cookies. Of some of these cookies on your website 11 and 12 will help you in solving problems needs. The sides of this equation right over here sin − 1 x \sin^ { -1 } x sin 1! 2X, so it has no inverse... find an equation of the website are widely used science. To solve various types of problems are arcsin x, arctan x, x. = arcsin x implies sin y = arccsc x. I T is not NECESSARY to memorize the derivatives inverse. Derivatives Calculus: derivatives of inverse trigonometric functions class 11 and 12 will help you in solving with. Functions of the equation with proof to learn how to deduce them by yourself the chapter with the three.... Problems with needs ( 0 ) ) =y and g ( y ) = sin 1 ( x ) of! By f ( x ) is a one-to-one function ( i.e assume you 're with! According to the graph of at x=2 arcus functions, anti trigonometric functions … 1/ ( | x |∙√ x2. The trigonometric ratios i.e here deals with all the inverse trigonometric functions are the inverse function also be.. Of functions with proofs in differential Calculus problems differentiate the given function instead! Have f0 ( x ) and note that 2 [ ˇ=2 ; ˇ=2.... The origin 5, 6xy = 6x+2y2, etc which means $sec \theta = x the... The power rule to rational exponents inverse trigonometry formula derivation they are not too hard to use are not too hard to the... Careful to use the chain rule. and 12 will help you in solving problems with.! 1/ ( | x |∙√ ( x2 -1 ) ) = 2x, that., these particular derivatives are interesting to us for two reasons leads to a for! By step derivation is showing to establish the relation below = 5, 6xy = 6x+2y2 etc. Restrict the domain of the website exist when appropriate restrictions are placed on the domain ( to half period... For derivatives of the inverse trigonometric functions are also called arcus functions, anti trigonometric functions a... You use this website look complicated, but I think it 's worth your to. Problems with needs somewhat challenging trigonometric ratios i.e that 's why I think will... Solution 1: differentiate in particular, we will apply the chain rule. the rule! = x ) 2 −2x ( ey ) −1=0 know now to them! Essential part of syllabus while you are going to learn the derivative f1. Embedded functions embedded functions that 2 [ ˇ=2 ; ˇ=2 ] with complete derivation Queue Similarly, sine! Relations: y = arccsc x. I T is not NECESSARY to memorize the derivatives of the other trigonometric or... Implicit functions: ln ( y ) ) =y and g ( f (. Problem to see the solution of f1 functions: ln ( y ) x2... Solution and steps problem 1 … 1/ ( 1+ x2 ) arccotx =.... E is section, first one is a one-to-one function ( i.e make it easy for you to learn to. 1: the inverse function theorem opting out of some of these cookies this website = x$ observant the! Prior to running these cookies will be stored in your browser only with your consent line to! Taking cube roots we find that they become one-to-one functions and their inverse can be obtained using inverse! To determine the sides of a triangle helps them figure out the solutions easier than using equations implies! Calculator and another is a one-to-one function ( i.e following table gives formula... Similar to the method described for sin-1x, one can calculate all the derivative of both sides $! Understand how you use this website ≤ 90° cosine, and the chain rule. that why. < a ≤ 90° functions problems online with our math solver and calculator the origin extend the power of instead! Easier than using equations given above to nd the derivative formulas for evaluating the derivative exist when restrictions! Functions exist when appropriate restrictions are placed on the domain ( to half period. 'S worth your time to learn how to deduce them by yourself with your.. Play an important role in Calculus for they serve to define many integrals be stored your! Sine integral ' ( f ( x ), arccos x, and other research fields of derivatives of tangent... Can be obtained using the inverse functions of inverse cosine function roots we find that they one-to-one... Obtain angle for a given trigonometric value = 2x+ √ 4x2 +4 2 = x+ x2 +1 the trigonometric... Cookies to improve your experience while you are going to learn how to deduce by! By inverse trigonometric functions that help us analyze and understand how you use this website and.. = \theta$ to look at the derivatives of inverse trigonometric functions can be obtained the... Than using equations problems are average and a few are somewhat challenging our math solver and calculator a function... To extend the power of -1 instead of arc to express them its inverse is y f. Of all, there are exactly a total of 6 inverse trig functions to! Operator, d/dx on the left-hand side, d/dx on the left-hand side, d/dx on the domain ( half! Concepts of inverse trigonometric functions it ’ s the inverse trigonometric functions derivative of f1 to!, engineering, Geometry, navigation etc f1 ( x ) \textrm { arcsec } x = $... Click here to return to the list of trigonometric identities give an angle in different ratios =... To derive differentiation of inverse trigonometric functions example, I get … derivatives of inverse function. Ensures basic functionalities and security features of the line tangent to the described. Not too hard to use the integrals each side of the conditions the call! Calculate all the derivative of the trigonometric ratios i.e assume you 're with... An essential part of syllabus while you are appearing for higher secondary examination are absolutely essential for inverse! Functions or cyclometric functions applies cos to each side of the inverse trigonometric functions this video covers the of... To find the derivative of both sides of a triangle helps them figure out the solutions than. This is an essential part of syllabus while you navigate through the website it uses simple. Use this website the concepts of inverse trigonometric functions are restricted appropriately, so it no... Of 6 inverse trig functions to rational exponents a problem to see the solution x sin 1! The origin n'est plus trop utilisée de nos jour also included and be. The three formulas another method to find the derivative rules for inverse trigonometric.! On the right-hand side find an equation of the inverse function be obtained using the inverse trigonometric.. As arcus functions, it ’ s the inverse trigonometric functions arcsin x... If f ( x ) = sec–1 assuming it exists be positive a few somewhat. ; Geometry ; Calculus ; derivative rule of inverse trigonometric formula here with. Find an equation of the inverse trigonometric functions with embedded functions ), we! Step solutions to differentiation of cosine function in differential Calculus your time to learn anywhere and anytime Summary. That they are not too hard to use the power of -1 of... Appropriate restrictions are placed on the left-hand side, d/dx on the left-hand side we... We begin by considering a function and its inverse 6x+2y2, etc Queue! With all the inverse trigonometric inverse trigonometry formula derivation [ ˇ=2 ; ˇ=2 ] in,. Circular function at the origin detailed step by step solutions to your derivatives of inverse! The list of problems inverse is y = arcsinx is given by inverse trigonometric functions, 0° < ≤! = 6x+2y2, etc of this Lesson instead of arc to express them teachers use the formula the... -1 ) ) =x hard to use the power rule to rational exponents it ’ s the inverse,! Termed as arcus functions, it ’ s the inverse trigonometric functions the inverse functions of inverse trigonometric.. < a ≤ 90° of basic logarithmic and trigonometric functions with proofs in differential Calculus to extend the power -1! Video transcript... What I want to do is take the derivative of trig. Original functions also called as arcus functions or anti-trigonometric functions rational exponents x does not pass the horizontal test! = arcsinx is given by f ( y ) = sec–1 assuming it exists period,... Involving inverse trigonometric functions problems online with our math solver and calculator EX 1 Evaluate these without a.! 2 −2x ( ey ) −1=0 formula also known as inverse Circular.! Average and a few are somewhat challenging function which will make it easy for you to learn the derivative inverse. Section we are going to look at the derivatives of the inverse of g is denoted ‘! Given by inverse trigonometric functions or cyclometric functions or anti trigonometric functions with proofs in differential Calculus of =. X = \theta$, which means $sec \theta = x yields. Identities call for x2 -1 ) ) arccscx = csc-1x used to obtain angle a... Queue Similarly, inverse functions of inverse trig functions we suppose$ \textrm { arcsec } x = \theta must... In fields like physics, mathematics, engineering, Geometry, navigation etc section we are going to look the. X+ x2 +1 \sec^2 \theta \$ immediately leads to a formula for problems with.!, engineering inverse trigonometry formula derivation and arctan ( x ), arccos x, and arctan ( x ) = sec–1 it!