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In the same way we may show that the formula for cos (a + B) is true for values of the angles as given above.

(2) Suppose that a is in the second quadrant and ß in the third,

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sin ẞ sin (180° + y)

sin (a + B) = sin [(90°

-sin y, cos B

- COS Y,

+ ø) + (180° + y)] = sin [270° + (ø + x)] cos o cos y + sin o sin y. Substituting for the functions of

·cos (p + y)

of the functions of a and ß,

sin (a + B)

=

and y their values in terms

- (sin a) (— cos ß) + (−cos a) (— sin ß) sin a cos ẞ+ cos a sin ß.

In the same manner it may be shown that the addition formulas are true for any angles.

69. Proof of the subtraction formulas for all values of the angles. Since the addition formulas are true for all values of a and ß, they are true when -ẞ is put for ẞ. Then

sin (a — ß) : sin [a + (−ẞ)] = sin a cos (— ß) + cos a sin (−ß), (−ẞ) sa

and

cos (a — B)

But sin (-a)

= cos [a + (−B)] = cos a cos (−ß) — sin a sin (—B).

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and

cos (a — ß) = cos a cos ẞ + sin a sin ẞ.

That is, the subtraction formulas are true in general.

EXERCISES

Prove that formulas [13] and [14] are true in the following cases:

1. a in the fourth quadrant and ẞ in the first.

2. a in the third quadrant and ẞ in the third.

3. a in the first quadrant and ẞ in the third.

Solve the following exercises by means of the addition and subtraction formulas, assuming a and ẞ less than 90°.

4. Find the sine and cosine of 90° by assuming that 90° Solution. sin 90°

=

sin (60° +30°)

=

=

60° + 30°. sin 60° cos 30° + cos 60° sin 30°.

Substituting the values of the functions of 30° and 60°,

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5

5. Given sin α = g and cosẞ; find sin (a +ẞ) and cos (a + B). Solution. Construct the right triangles ABC and DEF, Fig. 80, with a an acute angle of ▲ ABC, and ẞ an acute angle of ▲ DEF.

By [13], sin (a + B)

sin a cos ẞ + cos a sin ß.

Substituting the values for sin a, cos a, sin ß, and cos ß from the triangles, § · √1⁄2 + ‡ • 13 = 133 + 48 = 83.

sin (a + B)

=

13

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6

sin a sin ß

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=

6. Given sin α = 1, and cosẞ √2; find sin (a +ẞ) and cos (a + ß). 7. Find the sine and cosine of 75°, having given the functions of 45° and Ans. (√6+ √2), † (√6 − √2). 8. Given cos α = 1, and cosẞ = }; find sin (a+ß), cos (a+ß), sin (a—ß), and cos (aß).

30°.

6

6

4

Ans. ¿ (√3+2√2), ¿ (1 − 2 √6), † (√3 − 2 √2), ¿ (1 + 2 √6). 9. Show that sin (45° + 30°) # sin 45° + sin 30°.

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0.9659 sin (45°

0.70711, sin 30°
0.70711 +0.5
1.20711.

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= 0.5,
1.20711.

30°) # sin 45° + sin 30°.

Note. The symbol means is not equal to.

10. Show that sin (25° + 37°) # sin 25° + sin 37°.
11. Show that cos (35° + 28°) # cos 35° + cos 28°.
12. Prove from Fig. 78 that sin (a + ß) ✯ sin a + sin ß.

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Therefore sin a + sin ẞsin (a + ß).

13. Prove from Fig. 78 that cos (a + B) cos a + cos ß.

Note. A very common mistake made by beginners in trigonometry is to assume that sin (a + ß) = sin a + sin ß, etc. Exercises 10, 11, 12, and 13 are given for the purpose of impressing the student with the fact that such relations are not true.

In the following exercises, the angles may have any values. 14. Find the sin 120° by using 120°

15. Find cos 150° by using (a) 150°

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90° + 30°.

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120° + 30°, (b) 150° = 210° – 60°,

16. Find sin 240° and cos 240° by using (a) 240° = 210° +30°, (b) 240°

= 300° — 60°, etc.

17. Given sin α = 3 -, cos B second. Find sin (a + ß), cos (a

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12

651

=

-13, a in the third quadrant, and ẞ in the + ß), cos (a B), sin (a Ans. sin (a + B) 16 = 18. Given sin a §, a in the second quadrant, third quadrant. Find sin (a + ß), cos (a + ß), sin +ẞ), Ans. sin (a + B)

5

B). 18, cos (a + B) 88888 and tan ß 12, ẞ in the (a — ß), and cos (a -18, cos (a + B)

16

6

Find the value of 0 in the following exercises.
19. cos (20° + a) cos (20° — a) + sin (20° + a) sin (20° — α) = cos 0.

20. cos 50° cos (85° — a) — sin 50° sin (85° — a) = cos 0.

Ans. 0

B).

6 3

65

2α.

Ans. 0 135° α. 21. sin (90° + 1 ß) cos (90° — 1 ß) + cos (90° + 1⁄2 ß) sin (90° – 1⁄2 ß) sin 0. Ans. 0 = 180°.

22. cos (45° — x) cos (45° + x) — sin (45° — x) sin (45° + x) = cos 0.

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24. By means of [13], [14], [15], and [16] prove the following relations.

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Find the value of the following expressions, using only the principal values

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70. Formulas for the tangents of the sum and the difference of two angles. By [7], [13], and [14],

tan (a + B)

sin (a + B)
cos (a + B)

sin a cos ẞ + cos a sin ß

cos a cos ß — sin a sin ß

Dividing, both numerator and denominator by cos a cos ẞ, and

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Since formulas [13], [14], [15], and [16] are true for all values of a and ß, the formulas [17] and [18] are true in general.

EXERCISES

1. Find tan 75°, and tan 15° by means of 45° and 30°.

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3 , and sin ß

=

61

=

៖៖.

- §, a in the fourth and ß in the third quadrant, -24, tan (a−ẞ) = ∞.

3. If cos a = 3 find tan (a + ß) and tan (a – B). Ans. tan (a+B)

=

7

In the following problems use only the values of the angles < 90°.

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4. If α = Solution. if possible.

tan-1 and ẞ = tan-1 1, find (a + ß).

Take the function of (a + ß) which involves the given functions

tan a +tan B

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Suggestion. Let a = cos-1x and B ß tan-1 y.

Then B = α

Take

the tangent of each side of the equation and substitute for tan ɑ and tan ß their values in terms of x and y.

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71. Functions of an angle in terms of functions of half the angle. Since the formulas for the sum of two

for all values of a and ẞ they will be true when ß Then sin (a + B) = sin (a + a)

angles are true

= α.

sin a cos a + cos a sin a.

That is,

[19]

sin 2 a

2 sin a cos a.

This formula may be stated as follows:

The sine of any angle is equal to twice the product of the sine and cosine of the half angle.

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