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13. Determine the number of degrees into which the right angle must be divided in order that an angle of 30° may be measured by the number 40.

14. Shew that as the angle increases each of the trigonometrical functions changes sign whenever it passes through the value 0 or ∞, and for no other value.

FORMULE INVOLVING FUNCTIONS OF ONE

ANGLE.

Prove the following formula:

1. sin Asin {(2n+1) 180°- A} − sin {(2n + 1) 180° + A} = − sin {(2n+2) 180° — A}.

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4. tan A=-tan (180o - A)
— tan (180° — A) =— tan {(2n + 1) 180o — A},
=tan (180°+ A) = tan {(2n + 1) 180o + A}.

5. cosec A = cosec {(2n.+ 1) 180° — A}.

= sec (-A) = sec (n. 360° — A).

6. sec A sec (-A)

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12. From the equation sin (a — 0) = cos (a +0), find 0.

13. If tan 0=2, find the value of a cos 0 +b sin 0.

a

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14. Eliminate between the equations,

m = cosec ✪ — sin 0,

n = sec 0 cos 0.

15. Give a general formula for all values of 4 which satisfy

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π

1. Find the number of degrees, minutes and seconds, in the angles, the circular measures of which are ,1.5, 2, π+1, and 3.14 respectively.

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2. Reduce to the circular measure the following angles; 14°, 15°, 120°, 4800° and 700.

3. If two-thirds of a right angle be assumed as the angular unit, what will be the numerical value of an angle of 45°?

4. Determine the angular unit by the assumption of which the following equation would be numerically true,

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5. Find the circular measure of 1'.

6. It may be shewn, that if a very small angle 0 is exAs pressed in units of circular measurement, sin 0=0-nearly;

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what change must be made in the formula if is expressed in seconds?

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5. cot + tan = sec e cosec = 2 cosec 20.

6. (sin + cos 0)2 = 1 + sin 20..

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19. cos (0+) = sin2 + cos & cos (20+).

20. 2 (sin2 0 sin2

+ cos2 0 cos2 4) = 1 + cos 20 cos 20.

21. sin (0 +4) — sin 0 = sin ✪ — sin (0 — 4) – 4 sin 0 sin2 o.

22. sin ne = 2 sin n 10 cos 0 sin n 20.

23.

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--

cos (0 + 4) sin (0 − 6) + cos (4 +¥) sin (4 — ¥)

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+cos (y+x) sin (†− x) + cos (x+0) cos (x − 0) = 0.

24. sin (a - ẞ) + sin (ß − y) + sin (y — a)

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α-β
2

25. 1+cos (a - B) + cos (B − y) + cos (y — a)

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26. 4 sin (0-4) sin (m0—4) cos (1—m) 0=1+cos 2 (1 —m) 0

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1

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27. 4 cos a cos ẞ cos y = cos (a− B − y) + cos (B — y − a)

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31. If tan sin 20, find the value of chd 0.

=

32. If A, B, C, be three angles the values of which form an arithmetical progression, then

sin A – sin C: cos C-cos A :: cos B : sin B,

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arithmetical mean between a and B, (a, ß, y, being each less

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35. Given sin 0 =m sin &, and tan 0 = n tan 4, find sin 0 and cos p.

36. Given sin (0 + a) = m sin 0, find 0 in terms adapted to logarithmic computation.

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2 sin ◊ sin then tan, tan, tan↓ are sin (0+)

in harmonical progression.

38. If 0++ √ = (2n + 1) 90o,

tan tan tan y tan 0 + tan 0 tan & = 1,

and sin 20+ sin 24 + sin 24 = ± 4 cos 0 cos & cos y.

39. The tangents of two angles are and

find the tangent of their sum.

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respectively,

G. P.

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