In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse ), and the cosine is
Concept: Trigonometric Ratios for Allied Angles: sin (-θ) = -sin θ. cos (-θ) = cos θ. sin (nπ + θ) = (-1) Reciprocal Functions: The inverse trigonometric formula of inverse sine, inverse cosine, and inverse tangent can also be expressed in the following forms. Sin-1 x = Cosec-1 1/x; Cos-1 x = Sec-1 1/x; Tan-1 x = Cot-1 1/x; Complementary Functions: The complementary functions of sine-cosine, tangent-cotangent, secant-cosecant, sum up to π/2. Sin-1 Explanation: sin−1( − x) = − sin−1x. Let y = − sin−1x. −y = sin−1x. sin( − y) = x by the inverse property. −siny = x by the odd function property sin( −x) = −sinx. Solved Examples for You. Question 1: Find the derivative of tan-1(sin-12x). Answer : We just need to apply the chain rule of differentiation and use the formulae for the derivatives of inverse trigonometric functions directly to solve this problem. Then one can get: d dx(tan−1(sin−12x)) = 1 1 + (sin−12x)2. 1 1– (2x)2− −−−−− The integration of cosine inverse is of the form. I = ∫cos–1xdx I = ∫ cos – 1 x d x. When using integration by parts it must have at least two functions, however this has only one function: cos–1x cos – 1 x. So consider the second function as 1 1. Now the integration becomes. I = ∫cos–1x ⋅ 1dx – – – (i) I = ∫ cos – 1