The circular functions

Excerpts

• The circular functions

  The cosine and sine are continuous for all values of $y$, the tangent except at the points where its definition fails.

• The circular functions

  The formulae for differentiation of the circular functions may now be deduced in the ordinary way, and the power series derived from Taylor’s Theorem.

• The circular functions

  The equation (1) defines a unique value of $y$ corresponding to every real value of $x$. As $y$ is continuous and strictly increasing, there is an inverse function $x = x(y)$, also continuous and steadily increasing.

• The circular functions

  We have thus defined $\cos y$ and $\sin y$ for all values of $y$, and $\tan y$ for all values of $y$ other than odd multiples of $\frac{1}{2}\pi$. The cosine and sine are continuous for all values of $y$, the tangent except at the points where its definition fails.

• The circular functions

  The further development of the theory depends merely on the addition formulae.

• The circular functions

  To determine the sign put $y_{2} = 0$. The equation reduces to $\cos y_{1} = ±\cos y_{1}$, which shows that the positive sign must be chosen for at least one value of $y_{2}$, viz. $y_{2} = 0$. It follows from considerations of continuity that the positive sign must be chosen in all cases.

• The circular functions

  An alternative theory of the circular functions is based on the theory of infinite series.

Equations

• The circular functions

  y = y(x) = \arctan x = \ds\int_{0}^{x} \frac{dt}{1 + t^{2}}

  Defines y as the arctangent of x, given as the integral of 1/(1+t^2) from 0 to x.

• The circular functions

  x = x(y) = \tan y

  Defines the tangent of y as the inverse function of the arctangent, so x equals tan y.

• The circular functions

  \frac{1}{2}\pi = \ds\int_{0}^{\infty} \frac{dt}{1 + t^{2}}

  Defines pi as twice the integral of 1/(1+t^2) from 0 to infinity.

• The circular functions

  \cos y = \dfrac{1}{\sqrt{1 + x^{2}}}

  Defines the cosine of y in terms of x, with the square root taken positive.

• The circular functions

  \sin y = \dfrac{x}{\sqrt{1 + x^{2}}}

  Defines the sine of y in terms of x, with the square root taken positive.

• The circular functions

  \tan(y + \pi) &= &&\tan y

  Extends the tangent beyond the basic interval by making it periodic with period pi.

• The circular functions

  \cos(y + \pi) &= -&&\cos y

  Extends the cosine beyond the basic interval so that shifting y by pi changes its sign.

• The circular functions

  \sin(y + \pi) &= -&&\sin y

  Extends the sine beyond the basic interval so that shifting y by pi changes its sign.

• The circular functions

  \tan (y_{1} + y_{2}) = \dfrac{\tan y_{1} + \tan y_{2}}{1 - \tan y_{1}\tan y_{2}}

  The tangent of a sum of two angles equals the sum of their tangents divided by one minus their product.

• The circular functions

  \cos(y_{1} + y_{2}) = ±(\cos y_{1}\cos y_{2} - \sin y_{1}\sin y_{2})

  The cosine of a sum of two angles, with the sign of the right side fixed as positive by a continuity argument.

• The circular functions

  \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots

  In the alternative infinite-series theory, cos x is defined by this power series.

Problems

No exercises in this chapter.