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NCERT Solutions for Class 10 Maths Chapter 8 Exercise for

These ncert book chapter wise questions and answers are very helpful fref CBSE board exam. Express the trigonometric ratios and in terms of. By using identity. Write cg other trigonometric ratios of A in terms of. By using identity. Prove the following identities, where the angles involved are acute angles for which the expressions are defined:. Dividing all terms by. Dividing all the terms by.

Ncert solution class ch 8 maths class 10 ex 8.4 free Maths includes text book solutions from Mathematics Book. Save my name, email, and website in this browser for the next time I comment. Download Now. Express the trigonometric ratios and in terms of Ans.

Write the other trigonometric ratios of A in terms of Ans. Evaluate: i ii Ans. Choose the correct option. S iii L. S viii L. Worst experience, the answers doesnot appear.

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Order my paper. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root as noted below , and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration.

To find x :. If a is positive, the convergence is quadratic , which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration.

This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used. The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n -digit numbers.

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers , that does contain solutions to the square root of a negative number.

The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principal square root function is holomorphic everywhere except on the set of non-positive real numbers on strictly negative reals it isn't even continuous. The above can also be expressed in terms of trigonometric functions :.

When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root: [21] [22]. In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative. Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general.

A similar problem appears with other complex functions with branch cuts, e. The third equality cannot be justified see invalid proof. Abel�Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of n th roots. In general matrices may have multiple square roots or even an infinitude of them. Each element of an integral domain has no more than 2 square roots.

The only square root of 0 in an integral domain is 0 itself. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.

A non-zero element of the field F q with q elements is a quadratic residue if it has a square root in F q. Otherwise, it is a quadratic non-residue. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory. Unlike in an integral domain, a square root in an arbitrary unital ring need not be unique up to sign. A square root of 0 is either 0 or a zero divisor.

Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. A square root can be constructed with a compass and straightedge. In his Elements , Euclid fl.

However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid. Euclid's second proof in Book VI depends on the theory of similar triangles. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle Class 9 Maths Chapter 2 Question Answer Free and denote the length CH as h.

Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above. From Wikipedia, the free encyclopedia. Number whose square is a given number. For other uses, see Square Roots disambiguation.

Main article: Methods of computing square roots. Using the Riemann surface of the square root, it is shown how the two leaves fit together.

Main article: Square root of a matrix. See also: Square root of a 2 by 2 matrix. Math Vault. Retrieved ISBN Archived from the original on Retrieved 19 January Mathematics: A Concise History and Philosophy.

New York: Springer-Verlag. Cambridge University Press. Archived from the original on 24 November Archived PDF from the original on The Life of Numbers. Introduction to Applied Mathematics for Environmental Science. Mathematics for Biological Scientists. Garland Science. A History of Greek Mathematics, Vol. Oxford: Clarendon Press. Elementary functions: algorithms and implementation. Archived from the original on November 22,

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