Suggestion. Expanding sin (A + x) and factoring,

If the numerical values of m and A are known, the numerical value of cot x becomes known and the angle x can be found in degrees by referring to the tables of trigonometric functions.

12. From sin (A - x) = m sin x find x.

13. From the equation tan (A + x) of A and m.

Suggestion.

The given equation may be written

tan (A + x)

tan x

whence by composition and division

=m tan x find x in terms

If the numerical values of m and A are known, the numerical value of sin (24 + x) becomes known and the angle 24 + x can be taken from the tables. Since A is given the angle x becomes known.

of m, n and

14. Find x from tan (A + x) tan x = m.

15. From the equation m cos x + n sin x = q

find x in terms

If the numerical values of m, n and q are known, the numerical values of k and y can be found. Then the numerical value of sin (+x) can be found and +x becomes known. are known x can be found.

+x and

When

29. In the formulas for sin (A+B), cos (A+B), tan (A+B), and cot (A+B) place B = A. The formulas become

sin (2A) = 2 sin A cos A ;

(12) cos (2A) = cos2 A sin2 A 2 cos2 A

=

. 1

place B = 2A. There results

sin (3A) = sin A cos (24) + cos A sin (24)

= sin A (1 - 2 sin2 A) + cos A (2 sin A cos A)

= sin A - 2 sin3 A + 2 sin A (1 — sin2 A)

11. In a right triangle whose sides are 3, 4 and 5 a straight line is drawn from the vertex of the right angle to the middle point of the hypotenuse. Find the angle which this line makes

with the hypotenuse.

Art. XII. Functions of Half Angles.

30. In the formula

cos (24) 12 sin' A place AB.

=

There results cos B12 sin2 (B). Solving for sin B,

(a)

The Sum and Difference of Functions
Expressed as a Product.

sin (A + B) = sin A cos B + cos A sin B

sin (A — B) sin A cos B — cos A sin B

=

sin (A + B) + sin (A – B) = 2 sin A cos B.

Now place A + B = x, A − By, whence A = and B(xy). (xy). Equation (a) becomes

(14) sin x + sin y = 2 sin (x + y) cos(x − y),

· 1 (x + y)

that is the sum of the sines of two angles is twice the product of the sine of one-half the sum of the angles and the cosine of onehalf the difference of the angles.

9. sin x sin y = {cos (xy) cos (x + y)}.

10. From the equation sin (A + x) sin x = m find x in terms of A and m.

Suggestion.

sin (A + x) sin x = {cos A - cos (A + 2x)}

= m.

Whence cos (A + 2x) = cos A −2m, which determines A+2x and x becomes known.

11. From cos (A

x) cos x = m find x in terms of m and A.

12. From cos (A + x) sin x = m find x in terms of m and A.