Plane trigonometry with practical applications, by Leonard E. Dickson.

Ch. X]

FUNCTIONS OF SEVERAL ANGLES

145

Since 22~~ is an acute angle, its functions are positive and
sin 22~~ =2 /2 —2, cos 22~~=4\2+/ 2.
EXERCISES ON FUNCTIONS OF HALF ANGLES
1. Given cos 30~= =-3, find sin 15~, cos 15~, tan 15~ in terms of
square roots.
2. Given cos 150~= -= 33, find sin 75~, cos 75~, tan 75~ exactly.
3. Find the sine and cosine of 72~.
Prove the identities
x 1-cos x x 1+ cos x
4. tanX - os 5. cot-=+cs
2 sin x 2 sin x
6. 1 + tan x tan I x = sec x. 7. csc x - cot x = tan x.
8. sec A + tan A = tan (45~ + I A).
9. 1 + cot A cot l A = csc A cot ~ A.
cos x l+tan x_ lq+sin x+-cos x
10.. cot I x.. 1-sin x 1-tanx 1l-sin x-cos x
12. tan A + 2sin2 A cotA = sin A.
13. A balloon rose vertically at a point whose horizontal distance is
2400 yards from an observer who found the angle of elevation to be 15~
when he first sighted the balloon, and 30~ at a later time. Find an exact
expression for the distance the balloon rose between the two observations.
14.* From the values of 1 + cos A and 1 - cos A in Art. 91, prove
that
A- s(s-a) s - /A (s-b)(s -c) t (s-b)(s-c)
cos~A== bc sin~A= - - C, tan~A =C)- be s be tbc s (s - a)
where s = ~ (a + b + c).
15.* Express the six functions of x in terms of t = tan I x.
16.* Given the functions of 18~ (Ex. 38 of Art. 101), how can we find
the functions of 12~, 6~, 3~?
104. Sum or difference of two sines or two cosines expressed
as a product.' We employ formulas (1) and (9), viz.,
sin (A + B) = sin A cos B + cos A sin B,
sin (A - B) = sin A cos B - cos A sin B.
1 The instructor who omits Art. 104 should omit Example 2 and Exercises 23-30
of Art. 105.