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The trigonometric function are periodic functions, and their primitive period is 2π for the sine and the cosine, and π for the tangent, which is increasing in each open interval (π/2 + kπ, π/2 + (k + 1)π). At each end point of these intervals, the tangent function has a vertical asymptote .
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
A right triangle with sides relative to an angle at the point. Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that.
Sec-1, SEC-1, sec-1, or sec −1 may refer to: . sec x−1 = sec(x)−1 = exsec(x) or exsecant of x, an old trigonometric function; sec −1 y = sec −1 (y), sometimes interpreted as arcsec(y) or arcsecant of y, the compositional inverse of the trigonometric function secant (see below for ambiguity)
Similar right triangles illustrating the tangent and secant trigonometric functions. Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ. The identities
Calculus. In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer.
Pythagorean identities. Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of by ; for the second, divide by . Similarly. Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle diagram above.
The external secant function ( exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function: It was introduced in 1855 by American civil engineer Charles Haslett, who used it in conjunction with the existing versine function, for designing and measuring circular sections of railroad track. [2]