CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

Excerpts

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  When one magnitude or ratio is so connected with another that the former changes with the latter, but is determinable for any given value of the latter, the former is said to be a function of the latter.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  The side of the hexagon inscribed is $= R$, and it subtends $\dfrac{\pi}{3}$ or $60°$.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  It should be observed that, while the number $\theta$ continuously increases, the numbers $\sin\theta$, $\cos\theta$ pass through a series of values between $+1$ and $-1$, and return to the same values again for every increase of $2\pi$ in the value of $\theta$.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  If $AF$ be measured towards $T$, it is to be considered $+$, and $-$, if in the contrary direction towards $T'$.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  Since the perpendicular from the centre of a circle on any chord bisects it at right angles, the ratio of the chord to the radius is twice the sine of half the angle subtended by the chord.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  The seven ratios defined above are altogether independent of the size of the circle described, and depend only on the angle.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  From these equations all the functions can be found, when one has been given.

Equations

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin\theta = \dfrac{BD}{R}

  The sine of the angle AOB is the perpendicular BD from the end of the arc onto the initial line OA, divided by the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cos\theta = \sin\left(\frac{\pi}{2} - \theta\right).

  The cosine of an angle is the sine of its complement.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin\theta = \cos\left(\frac{\pi}{2} - \theta\right).

  The sine of an angle is the cosine of its complement.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cos\theta = \sin COB = \frac{BE}{R} = \frac{OD}{R}.

  The cosine of the angle AOB equals the sine of its complement COB, which in the diagram is the ratio OD to the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin\theta &= \sin(2m\pi + \theta);

  Adding a whole number multiple of 2π to the angle leaves the sine unchanged.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cos\theta &= \cos(2m\pi + \theta).

  Adding a whole number multiple of 2π to the angle leaves the cosine unchanged.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin(-\theta) &= -\sin\theta;

  The sine of the negative of an angle is the negative of its sine.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cos(-\theta) &= +\cos\theta.

  The cosine of the negative of an angle equals the cosine of the angle.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cos(\pi - \theta) &= -\cos\theta;

  The cosine of π minus an angle is the negative of the cosine of the angle.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \tan\theta = \frac{AF}{R} \text{ and } AF = R\tan\theta.

  The tangent of an angle is the length AF cut off on the tangent at the initial end of the arc, divided by the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \tan(-\theta) = -\tan\theta; \text{ and } \cot(-\theta) = -\cot\theta.

  The tangent and cotangent of the negative of an angle are the negatives of those of the angle.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \tan(\pi - \theta) = -\tan\theta; \quad \cot(\pi - \theta) = -\cot\theta.

  For supplementary angles the tangent and cotangent are equal in size and opposite in sign.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cot\theta = \frac{CG}{R} \text{ and } CG = R\cot\theta.

  The cotangent of an angle is the length CG cut off on the tangent at C to the arc of the complement, divided by the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sec\theta = \frac{OS}{R};\quad \cosec\theta = \frac{OK}{R};

  The secant and cosecant of an angle are the lengths OS and OK, cut off by the tangent at B, divided by the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \versin\theta = \frac{AD}{R}; \text{ and } AD = R\versin\theta.

  The versed sine of an angle is the segment AD cut off on the initial radius by the perpendicular from B, divided by the radius.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  R^2 = R^2 \sin^2\theta + R^2 \cos^2\theta,

  The square on the radius equals the sum of the squares on the sine and cosine lines, by the Pythagorean theorem applied to the right triangle BOD.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin^2\theta + \cos^2\theta = 1\Add{.}

  The square of the sine plus the square of the cosine of any angle equals one.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sec^2\theta = 1 + \tan^2\theta\Add{.}

  The square of the secant equals one plus the square of the tangent.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cosec^2\theta = 1 + \cot^2\theta\Add{.}

  The square of the cosecant equals one plus the square of the cotangent.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \tan\theta = \frac{\sin\theta}{\cos\theta}

  The tangent of an angle is its sine divided by its cosine.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sec\theta = \frac{1}{\cos\theta}\Add{.}

  The secant of an angle is the reciprocal of its cosine.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cotan\theta = \frac{\cos\theta}{\sin\theta} &= \frac{1}{\tan\theta}

  The cotangent of an angle is its cosine over its sine, and the reciprocal of its tangent.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \cosec\theta &= \frac{1}{\sin\theta}

  The cosecant of an angle is the reciprocal of its sine.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \versin\theta = 1 - \cos\theta\Add{.}

  The versed sine of an angle equals one minus its cosine.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin \frac{\pi}{4} = \sin 45° = \frac{1}{2}\sqrt{2} = \cos 45° = \cos \frac{\pi}{4}.

  The sine and cosine of 45° (π/4) both equal √2/2, from the side of the inscribed square.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin \frac{\pi}{6} = \sin 30° = \frac{1}{2} = \cos 60° = \cos \frac{\pi}{3}.

  The sine of 30° (π/6) equals one half, from the side of the inscribed hexagon.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin \frac{\pi}{3} = \sin 60° = \frac{1}{2}\sqrt{3} = \cos 30° = \cos \frac{\pi}{6}.

  The sine of 60° (π/3) equals √3/2, from the side of the inscribed equilateral triangle.

• CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS

  \sin \frac{\pi}{10} = \sin 18° = \dfrac{\sqrt{5} - 1}{4} = \cos \frac{4\pi}{10} = \cos 72°.

  The sine of 18° (π/10) equals (√5 − 1)/4, from the side of the inscribed regular decagon.

Problems

No exercises in this chapter.