Plane Trigonometry

Trigonometrical Ratios

606 Let 𝑂𝐴 be fixed, and let the revolving line 𝑂𝑃 describe a circle round 𝑂. Draw 𝑃𝑁 always perpendicular to 𝐴𝐴'. Then, in all positions of 𝑂𝑃.
𝑃𝑁𝑂𝑃= the sine of the angle 𝐴𝑂𝑃. 𝑂𝑁𝑂𝑃= the cosine of the angle 𝐴𝑂𝑃. 𝑃𝑁𝑂𝑁= the tangent of the angle 𝐴𝑂𝑃. 607 𝑃𝑁 always perpendicular to 𝐴𝐴'. Then, in all positions of 𝑂𝑃.
If 𝑃 be above the line 𝐴𝐴', sin 𝐴𝑂𝑃 is positive.
If 𝑃 be below the line 𝐴𝐴', sin 𝐴𝑂𝑃 is negative. 608 If 𝑃 lies to the right of BB', cos 𝐴𝑂𝑃 is positive.
If 𝑃 lies to the left of BB', cos 𝐴𝑂𝑃 is negative. 609 Note, that by the angle 𝐴𝑂𝑃 is meant the angle through which 𝑂𝑃 has revolved from 𝑂𝐴, its initial position; and this angle of revolution may have any magnitude. If the revolution takes place in the opposite direction, the angle described is reckoned negative. 610 The secant of an angle is the reciprocal of its cosine, or cos 𝐴sec𝐴=1 611 The cosecant of an angle is the reciprocal of its sine, or sin 𝐴cosec 𝐴=1 612 The cotangent of an angle is the reciprocal of its tangent, or tan 𝐴cot𝐴=1

Relations between the trigonometrical functions of the same angle.

613 sin²𝐴cos²𝐴=11.47 614 sec²𝐴=1+tan²𝐴 615 cosec²𝐴=1+cot²𝐴 616 tan 𝐴=sin 𝐴cos 𝐴606 617 If tan 𝐴=𝑎𝑏, sin 𝐴=𝑎𝑎²+𝑏² cos 𝐴=𝑏𝑎²+𝑏²606 618 sin 𝐴=tan 𝐴1+tan²𝐴; cos 𝐴=11+tan²𝐴617 619 The Complement of 𝐴 is = 90°−𝐴 620 The Supplement of 𝐴 is = 180°−𝐴 621 sin(90°−𝐴)=cos 𝐴 tan(90°−𝐴)=cot 𝐴 sec(90°−𝐴)=cosec 𝐴 622 sin(180°−𝐴)=sin 𝐴 cos(180°−𝐴)=−cos 𝐴 tan(180°−𝐴)=−tan 𝐴 In the figure ∠𝑄O𝑋=180°−𝐴607, 608 623 sin(−𝐴)=−sin 𝐴 624 cos(−𝐴)=cos 𝐴By Fig., and 607, 608 The secant, cosecant, and cotangent of 180°−𝐴, and of −𝐴, will follow the same rule as their reciprocals, the cosine, sine, and tangent.610, 612 625 To reduce any ratio of an angle greater than 90° to the ratio of an angle less than 90°. Rule: Determine the sign of the ratio by the rules (607), and then substitute for the given angle the acute angle formed by its tow bounding lines, produced if necessary.

Example

To find all the ratios of 660°. Measuring 300° (=660°−360°) round the circle from 𝐴 to 𝑃, we find the acute angle 𝐴𝑂𝑃 to be 60° and 𝑃 lies below 𝐴𝐴′, and to the right of 𝐵𝐵′.
Therefore sin 660°=−sin 60°=−√32 cos 660°=cos 60°=12 and from the sine and cosine all the remaining ratios may be found by (610-616). 626

Inverse Notation

The angle whose sine is 𝑥 is denoted by sin⁻¹𝑥.
All the angles which have a given sine, cosine, or tangent, are given by the formula, sin⁻¹𝑥=𝑛𝜋+(−1)^𝑛𝜃1 cos⁻¹𝑥=2𝑛𝜋±𝜃2 tan⁻¹𝑥=𝑛𝜋+𝜃3 In these formula 𝜃 is any angle which has 𝑥 for its sine, cosine, or tangent respectively, and 𝑛 is any integer.
cosec⁻¹𝑥, sec⁻¹𝑥, cot⁻¹𝑥 have similar general values, by (610-612).
These formula are verified by taking 𝐴, in Fig. 622, for 𝜃, and making 𝑛 an odd or even integer successively.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive