THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

Excerpts

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  The four expressions for $\sin (\theta ± \phi)$ and $\cos (\theta ± \phi)$ are true for all values of $\theta$ and $\phi$, though the diagrams suppose the angles all acute. They form the fundamental formulæ of Analytical Trigonometry.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  Then $\theta + \phi$ is the circular measure of $ACD$.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  Then $\theta - \phi =$ circular measure of $ACD$.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  To shew that, when the difference between $R$ and $r$ is small, the limit of each radius is very nearly $= \dfrac{1}{3}(r + 2R)$.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  So that when, as in the text, $R - r < .00003$ the error in taking the ultimate radius $= \dfrac{1}{3}(r + 2R)$ is $< \dfrac{.0000000009}{45r}$ which does not affect the tenth decimal place.

Equations

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  DE &= R \sin (\theta + \phi)

  The perpendicular DE from D to CA equals R times the sine of the sum of the two angles.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \sin (\theta + \phi) &= \sin\theta \cos\phi + \cos\theta \sin\phi

  The sine of the sum of two angles is the sine of the first times the cosine of the second plus the cosine of the first times the sine of the second.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \cos (\theta + \phi) &= \cos\theta \cos\phi - \sin\theta \sin\phi

  The cosine of the sum of two angles is the product of the cosines minus the product of the sines.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \sin (\theta - \phi) &= \sin\theta \cos\phi - \cos\theta \sin\phi

  The sine of the difference of two angles is the sine of the first times the cosine of the second minus the cosine of the first times the sine of the second.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \cos (\theta - \phi) &= \cos\phi \cos\theta + \sin\phi \sin\theta

  The cosine of the difference of two angles is the product of the cosines plus the product of the sines.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  R - r = 2\delta

  The difference between the two radii R and r is written as twice the quantity δ.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  r_1 = \dfrac{R + r}{2}

  The inscribed radius of the first polygon is the average of R and r.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  R_1^2 &= r_1R

  The square of the circumscribed radius of the first doubled polygon equals the product of r_1 and R.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \delta_1 = \frac{\delta}{4} - \frac{\delta_1^2}{r_1}

  The first correction δ_1 equals one quarter of δ less a small quadratic term in δ_1 divided by r_1.

• THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES

  \delta + \delta_1 + \delta_2 + \dots < \frac{4}{3}\, \delta

  The infinite sum of the successive corrections is less than four-thirds of δ.

Problems

No exercises in this chapter.