Sin inverse x formula

Table of Contents • • • What is the Derivative of Sin Inverse x? (Substitution Method) Step 1: Let y=sin -1 x Step 2: Applying the sine function on both sides, we have sin y =sin sin -1 x ⇒ sin y =x $\cdots (i)$ Step 3: Differentiating with respect to x, we get $\cos y \dfrac$ BACK to HOME PAGE

Inverse Sine The inverse sine function (also called arcsine) is the inverse of sine function. Since sine of an angle ( sine function ) is equal to ratio of opposite side and hypotenuse, thus sine inverse of same ratio will give the measure of the angle. Let’s say θ is the angle, then: sin θ = (Opposite side to θ/Hypotenuse) Or θ = Sin -1  (Opposite side to θ/Hypotenuse) In the same way, Inverse Cosine and Inverse Tan can be determined. Fact : If the ratio of opposite side of angle θ and hypotenuse equals to 1, then, θ = sin -1  (1) = sin -1  (sin 90) = 90 degrees The function of inverse or inverse function is used to determine the angle measure, with the use of basic trigonometric ratios from the right triangle. Mostly, the inverse sine function is represented using sin -1. It does not mean that sine is not raised to the negative power. Table of contents: • • • • • • What is Sine Function? In a right-angle triangle, a sine function of an angle θ is equal to the opposite side to θ divided by hypotenuse. Sin θ = Opposite side/Hypotenuse This is the basic formula for sine function. See the figure below. Sin θ = Opposite / Hypotenuse What is Inverse Sine Function? In the above explanation, we learned about sine function, where we can determine the sin of any angle if the opposite side and hypotenuse is known to us.  What if we have to find just the measure of angle θ? The inverse sine function or Sin -1   takes the ratio, Opposite Side / Hypotenuse Side a...

Arcsine function arcsin(x), sin -1(x), inverse • • • • • The arcsine of x is defined as the inverse When the sine of y is equal to x: sin y = x Then the arcsine of x is equal to the inverse sine function of x, which is equal to y: arcsin x = sin -1 x = y Example arcsin 1 = sin -1 1 = π/2 rad = 90° Rule name Rule sin( arcsin x ) = x arcsin( sin x ) = x+2 kπ, when k∈ℤ ( k is integer) Arcsin of negative argument arcsin(- x) = - arcsin x Complementary angles arcsin x = π/2 - arccos x = 90° - arccos x Arcsin sum arcsin α + arcsin( β) = arcsin( α√ (1- β 2) + β√ (1- α 2) ) Arcsin difference arcsin α - arcsin( β) = arcsin( α√ (1- β 2) - β√ (1- α 2) ) Indefinite integral of arcsine x arcsin(x) (rad) arcsin(x) (°) -1 -π/2 -90° -√ 3/2 -π/3 -60° -√ 2/2 -π/4 -45° -1/2 -π/6 -30° 0 0 0° 1/2 π/6 30° √ 2/2 π/4 45° √ 3/2 π/3 60° 1 π/2 90° See also • • • • • • • • • • • • • •

Sine function sin(x), sine function. • • • • • • Sine definition In a right triangle ABC the sine of α, sin(α) is defined as the ratio betwween the side opposite to angle α and the side opposite to the right angle (hypotenuse): sin α = a / c Example a = 3" c = 5" sin α = a / c = 3 / 5 = 0.6 Graph of sine TBD Sine rules Rule name Rule Symmetry sin(- θ) = -sin θ Symmetry sin(90° - θ) = cos θ Pythagorean identity sin 2 α + cos 2 α = 1 sin θ = cos θ× tan θ sin θ = 1 / csc θ Double angle sin 2 θ = 2 sin θ cos θ Angles sum sin( α+β) = sin α cos β + cos α sin β Angles difference sin( α-β) = sin αcos β - cos α sin β Sum to product sin α + sin β = 2 sin [( α+β)/2] cos [( α- β)/2] Difference to product sin α - sin β = 2 sin [( α-β)/2] cos [( α+β)/2] Law of sines a / sin α = b / sin β = c / sin γ Derivative sin' x = cos x Integral ∫ sin x d x = - cos x + C Euler's formula sin x = ( e ix - e - ix) / 2 i The When the sine of y is equal to x: sin y = x Then the arcsine of x is equal to the inverse sine function of x, which is equal to y: arcsin x = sin -1( x) = y See: Sine table x (°) x (rad) sin x -90° -π/2 -1 -60° -π/3 -√ 3/2 -45° -π/4 -√ 2/2 -30° -π/6 -1/2 0° 0 0 30° π/6 1/2 45° π/4 √ 2/2 60° π/3 √ 3/2 90° π/2 1 See also • • • •

sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure. In inverse hyperbolic functions are arc- or ar-. For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding arsinh ⁡ ( sinh ⁡ a ) = a Some authors call the inverse hyperbolic functions hyperbolic area functions. Hyperbolic functions occur in the calculations of angles and distances in Notation [ ] x 2 − y 2 = 1 If the argument of a square root is real, then z is real, and it follows that both principal values of square roots are defined, except if z is real and belongs to one of the intervals (−∞, 0] and [1, +∞). If the argument of the logarithm is real and negative, then z is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two (−∞, 0] and [1, +∞). For z = 0, there is a singular point that is included in one of the branch cuts. Graphical representation [ ] In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains. In other words, the above defined Durán, Mario (2012). Mathematical methods for ...

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\( \newcommand\) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Learning Objectives • Understand and use the inverse sine, cosine, and tangent functions. • Find the exact value of expressions involving the inverse sine, cosine, and tangent functions. • Use a calculator to evaluate inverse trigonometric functions. • Find exact values of composite functions with inverse trigonometric functions. For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions. Understanding and Using the Inverse Sine, Cosine, and Tangent Functions In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure \(\PageIndex\): The tangent function and inverse tangent (or arctangent) function RELATIONS FOR INVERSE SINE, COSINE, AND TANGENT FUNCTIONS For angles in the interval \(\left[ −\dfracx=y\). Example \(\PageIndex\) Finding the Exact Value of...

Hyperbolic Functions: Inverses Hyperbolic Functions: Inverses The hyperbolic sine function, \sinh x, is one-to-one, and therefore has a well-defined inverse, \sinh^\right).$$

\( \newcommand\) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Learning Objectives • Understand and use the inverse sine, cosine, and tangent functions. • Find the exact value of expressions involving the inverse sine, cosine, and tangent functions. • Use a calculator to evaluate inverse trigonometric functions. • Find exact values of composite functions with inverse trigonometric functions. For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions. Understanding and Using the Inverse Sine, Cosine, and Tangent Functions In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure \(\PageIndex\): The tangent function and inverse tangent (or arctangent) function RELATIONS FOR INVERSE SINE, COSINE, AND TANGENT FUNCTIONS For angles in the interval \(\left[ −\dfracx=y\). Example \(\PageIndex\) Finding the Exact Value o...

sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure. In inverse hyperbolic functions are arc- or ar-. For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding arsinh ⁡ ( sinh ⁡ a ) = a Some authors call the inverse hyperbolic functions hyperbolic area functions. Hyperbolic functions occur in the calculations of angles and distances in Notation [ ] x 2 − y 2 = 1 If the argument of a square root is real, then z is real, and it follows that both principal values of square roots are defined, except if z is real and belongs to one of the intervals (−∞, 0] and [1, +∞). If the argument of the logarithm is real and negative, then z is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two (−∞, 0] and [1, +∞). For z = 0, there is a singular point that is included in one of the branch cuts. Graphical representation [ ] In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains. In other words, the above defined Durán, Mario (2012). Mathematical methods for ...