James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix D Trigonometry A29
12a
12b
sinsx 1 yd − sin x cos y 1 cos x sin y
cossx 1 yd − cos x cos y 2 sin x sin y
The proofs of these addition formulas are outlined in Exercises 85, 86, and 87.
By substituting 2y for y in Equations 12a and 12b and using Equations 10a and 10b,
we obtain the following subtraction formulas:
13a
13b
sinsx 2 yd − sin x cos y 2 cos x sin y
cossx 2 yd − cos x cos y 1 sin x sin y
Then, by dividing the formulas in Equations 12 or Equations 13, we obtain the corresponding
formulas for tansx 6 yd:
14a tansx 1 yd −
14b tansx 2 yd −
tan x 1 tan y
1 2 tan x tan y
tan x 2 tan y
1 1 tan x tan y
If we put y − x in the addition formulas (12), we get the double-angle formulas:
15a
15b
sin 2x − 2 sin x cos x
cos 2x − cos 2 x 2 sin 2 x
Then, by using the identity sin 2 x 1 cos 2 x − 1, we obtain the following alternate forms
of the double-angle formulas for cos 2x:
16a cos 2x − 2 cos 2 x 2 1
16b
cos 2x − 1 2 2 sin 2 x
If we now solve these equations for cos 2 x and sin 2 x, we get the following half-angle
formulas, which are useful in integral calculus:
17a
17b
cos 2 x −
sin 2 x −
1 1 cos 2x
2
1 2 cos 2x
2
Finally, we state the product formulas, which can be deduced from Equations 12
and 13:
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