*CHAPTER V.

TRIGONOMETRICAL RATIOS OF KNOWN ANGLES.

FORMULAE

OF VERIFICATION AND FOR THE DIVISION OF ANGLES.

42. Find the principal Trigonometrical Ratios of the following angles: 30°, 60°, 45°, 15°, 75°, 36°, 54°, 18°, 72°, 9°, 81°, 27°, and 63°.

The Trigonometrical Ratios of all these angles will be positive, since the angles are positive and less than 90°.

sin 60° = 2 sin 30° cos 30°

(Art. 37)

= 2 sin 30° sin 60°,

since cos 30°=sin(90° - 30°)

.. 1=2sin 30°, .. sin 30° = cos 60° = 1⁄2,

(Art. 15);

.. cos 30° = sin 60° = √(1 − sin330°) = √√/(1 − 1) = {√3,

sin245° + cos345° = 1, and sin 45° = cos 45°,

sin 75° sin(45° +30°) = sin 45° cos 30° + cos 45° sin 30°

=

cos 75° = cos(45° +30°) = cos 45° cos 30°-sin 45° sin30°

Let A 18°, .. 3A = 54°, 24=36°, and 3A+2A = 90°,

=

.. cos 3A = sin 24,

3A+2A=90°,

.. 4 cos3A - 3 cosA = 2 sinA cosA (Arts. 39, 37),

Since sin 18° is positive we must take the positive root; .. sin 18° = cos 72° = (√5-1),

. ́. cos 18° = sin 72° = √(1 − sino18°) = √√(1-6-2/5)

.. sin 36° = √/(1 — cos136°) = √ √/ (16+25)

= √(10-2√√5) = cos 54°.

By Art. (33) we have

sin 54° + sin 36° = 2 sin 45° cos 9° = √√2. cos 9°,

sin 54° - sin 36° = 2 cos 45° sin 9° = √√2. sin 9°,

.. cos 9° = sin 81°

√/5+1, by the ordinary rule for extract

ing the square root of a Binomial Surd.

And

sin 9° = cos 81°

√/5+1−√(10−2√√5) _ √(3 + √ 5) −√(5−√5)

=

=

4√/2

4

√/2

cos 18+ sin 18° √(10+2√/5) + √√5 −1

4√/2

4√2

cos 18° - sin 18° (10+2√√5) −(√5—1)

√/2

The Trigonometrical Ratios of 30° and 60° may also be easily found by drawing a perpendicular from one angle of an equilateral triangle to the opposite side.

42. Reduce sin0+ cos 0 to a single term.

43. By Art. (33) we have

sin(36+4) sin(36° - A) = 2 cos 36° sinA =

sin(72°+A) — sin(72° — A)=2 cos 72° sinA =

(√5+1) sinA,

(√5—1) sinA,

.. sin(36°+A) + sin(72° — A) — sin(36° — A) — sin(72° + A) =(√5+1) sin4-(√5-1) sinA = sinA.

(Euler's Formula.)

sin(54°+A) + sin(54° — A) = 2 sin 54° cos▲= √(√5+1) cos▲,

sin(18°+A) +sin(18° — A) = 2 sin 18° cosД= (√5—1) cos▲, ... sin(54° + A) + sin (54° — A) — sin (18° + A) − sin(18° — A), = (√5+ 1) cosA – (√5 − 1) cosA = cosA.

(Legendre's Formula.)

sin(60°+ A) — sin(60° — A) = 2 cos 60 sinA = sinΑ,

sin(45°+ A) — sin (45° — A) = 2 cos 45° sinA = √2. sinA.

Euler's and Legendre's formulae are called formulae of verification, because they are used in verifying results in the computation of Trigonometrical Tables.

2

(sinA - cos)2 = 1 − sinД;

..sinД+cosA = ± √(1 + sinA))
sin-cos A = ± √(1 − sin A))
J... (1).

.. 2sin+A=±√(1 + sinA) ± √(1 − sinĀ))

2 cos4=±√(1+sinA) ✈ √(1 − sinA)

The ambiguity of sign before the radicals in (1) and (2)

may

be removed by the particular value or limits assigned to A, as will appear from the next Article.

45. To trace the change in the sign of sin A+ cos A and of sin A—cosA as A varies from 0° to 360°, that is, as A varies from 0° to 720°.

Let the dotted lines bisect the four quadrants.

It is clear that the sines and cosines of the angles terminated by the dotted lines are numerically equal, and that the numerically greater of the two ratios sin and cos4 will determine the signs of both expressions sinA + cos

sin 4-cos4.

and

Now for values of A from -45° to +45°, that is, through