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32. 300°.

33. -30°.


- 225°.

35. Given sinx- -√, and cosx negative; find the other functions of x, and the value of x.

36. Given cotx=-√3, and x in Quadrant II.; find the other functions of x, and the value of x.

37. Find the functions of 3540°.

38. What angles less than 360° have a sine equal to -? a tangent equal to −√3?

39. Which of the angles mentioned in Examples 27–34 have a cosine equal to -√? a cotangent equal to ~√3?

40. What values of x between 0° and 720° will satisfy the equation sinx + } ?

41. In each of the following cases find the other angle between 0° and 360° for which the corresponding function (sign included) has the same value: sin 12°, cos 26°, tan 45°, cot 72°; sin 1910, cos 120°, tan 244°, cot 357°.

42. Given tan 238° 1.6; find sin 122°.

= =

43. Given cos 333° = 0.89; find tan 117°.

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45. m cos (90° - x) sin (90° — x).

46. (a - b) tan (90° − x) + (a+b) cot (90°+x).

47. a2+b2-2 ab cos (180° — x).

48. sin (90°+x) sin (180°+x)+cos (90° +x) cos (180° — x). 49. cos (180°+x) cos (270°-y)-sin (180°+x) sin (270° —y). 50. tanx+tan (y) — tan (180° - y).

51. For what values of x is the expression sin x + cos x positive, and for what values negative? Represent the result by a drawing in which the sectors corresponding to the negative values are shaded.

52. Answer the question of last example for sin x-cosx. 53. Find the functions of (x — 90°) in terms of the functions of x.

54. Find the functions of (x-180°) in terms of the functions

of x.

§ 31. FUNCTIONS OF THE SUM OF TWO ANGLES. In a unit circle (Fig. 27) let the angle AOB = x, the angle BOC=y; then the angle AOC=


In order to express sin(x+y) and cos(x+y) in terms of the sines and cosines of x and y, draw CFLOA, CDL OB, DEL OA, DGL CF; then CD siny, OD =cosy, and the angle DCG= the angle GDO = x. Also, sin(x+y)= CF-DE+CG.






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sinx; hence, DE= sin x x OD = sin x cosy.

= cos x; hence, CG = cos x x CD = cos x sin y.



sin(x+y)= sinx cos y+cosx siny.

cos(x+y)=OF=OE- DG.

OE =cosx; hence, OE= cos x x OD = cos x cosy.


DG =


= sinx; hence, DG = sinx × CD = sin x siny.


cos(x+y) = cos x cos y - sin x sin y.



In this proof x and y, and also the sum x+y, are assumed


Fig. 28.


to be acute angles. If the sum x+y of the acute angles x and y is obtuse, as in Fig. 28, the proof remains, word for word, the same as above, the only dif ference being that the sign of OF will be negative, as DG is above formulas, therefore, hold

now greater than OE. The true for all acute angles x and y.

If these formulas hold true for any two acute angles x and y, they hold true when one of the angles is increased by 90°. Thus, if for x we write x'=90°+x, then, by § 29,

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cos(x+y)= cos(90°+x+y)=sin(x+y).


Hence, by [5], sin (x'+y)=


cosx cosy — sin x sing,

Now, by §29, cosx=

by [4], cos(x+y)=- sin x cosy - cos x sin y.

sin (90° +x) = sin x',·

sin x=- cos (90°+x) = — cosx'.

Therefore, by putting sin a' for cosx, and cos x for sinx, in the right-hand members of the above equations,

sin(2+)=sin 2' cosy + cos. ' sing,
cos(x'+y)=cosx' cosy - sin x' sin y.

Hence, it follows that Formulas [4] and [5] are universally true. For they have been proved true for any two acute angles, and also true when one of these angles is increased by 90°; hence they are true for each repeated increase of one or the other angle by 90°, and therefore true for the sum of any two angles whatever.

By § 27,

tan (x+y)


sin ( + ) _ sin x cosy + cosx sin
cos(x+y) cosx cosy — sin x sing

If we divide each term of the numerator and denominator of the last fraction by cosx cosy, and again apply § 27, we obtain

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In like manner, by dividing each term of the numerator and denominator of the value of cot (x+y) by sin x sin y, we obtain

cot (x+y)

cotx coty - 1


§32. FUNCTIONS OF THE DIFFERENCE OF TWO ANGLES. In a unit circle (Fig. 29) let the angle AOB=x, the angle COB = y; then the angle AOC=

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sin x; hence, DE= sin x x OD = sin x cosy.

=cosx; hence, CG =cosxx CD= cos x siny.

Therefore, sin(x-y) = sin x cos y cos x sin y.

Again, cos (x − y) = OF=OE+DG.



=cosx; hence, OE

= cos x X OD

= cos x cosy.




= sin x; hence, DG = sin x × CD = sin x sin y.

Therefore, cos(x-y) = cos x cos y + sin x sin y.


In this proof, both x and y are assumed to be acute angles; but, whatever be the values of x and y, the same method of proof will always lead to Formulas [8] and [9], when due regard is paid to the algebraic signs.

The general application of these formulas may be at once. shown by deducing them from the general formulas established in § 31, as follows:

It is obvious that (xy)+y=x. If we apply Formulas [4] and [5] to (x − y)+y, then

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sin{(x − y) + y} or sin x = sin (x − y) cosy + cos(x − y) sin y,

cos{(x − y)+y} or cos x = cos(x − y) cosy — sin (x − y) sin y.

Multiply the first equation by cosy, the second by siny,

sin x cosy =

cosx siny =

sin(xy) cos2y+cos(x-y) siny cosy,

- sin (x − y) sin2y + cos(x − y) sin y cosy;

whence, by subtraction,

sin x cosy - cosx sin y=sin(x − y) (sin3y + cos3y).



therefore, by transposing,

sin (x − y) = sin x cosy — cos x sin y.

Again, if we multiply the first equation by sin y, the second equation by cosy, and add the results, we obtain, by reducing,

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

Therefore, Formulas [8] and [9], like [4] and [5], from which they have been derived, are universally true.

From [8] and [9], by proceeding as in § 31, we obtain

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Formulas [4]-[11] may be combined as follows:

sin (x+y)= sin x cosy ± cos x siny,

cos (x + y) = cos x cos y sin x siny,

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