Writing The co in cosine comes from complementary,since the cosine of an angle is the

Chapter 2, Problem 84

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Writing The "co" in "cosine" comes from "complementary," since the cosine of an angle is the sine of the complementary angle, and vice versa:

 

                                 \(\cos x=\sin \left(\frac{\pi}{2}-x\right)\) and \(\sin x=\cos \left(\frac{\pi}{2}-x\right)\)

                           

Suppose that we define a function \(g\) to be a cofunction of a function \(f\) if

                                       \(g(x)=f\left(\frac{\pi}{2}-x\right)\) for all \(x\)

Thus, cosine and sine are cofunctions of each other, as are cotangent and tangent, and also cosecant and secant. If \(g\) is the cofunction of \(f\), state a formula that relates \(g^{\prime}\) and the cofunction of \(f^{\prime}\). Discuss how this relationship is exhibited by the derivatives of the cosine, cotangent, and cosecant functions.

Equation Transcription:

Text Transcription:

cos x=sin (pi/2-x)

sin x=cos(pi/2-x)

g

f

g(x)=f(pi/2-x)

x

g

f

g'

f'