19. Limiting values of the functions. The values

of a and o given in the preceding article, substituted in group (B) give

[The order of the signs ± indicates that the function has passed from + to; that is just before the generating line reached the given angle the sign of the function was, and was immediately after passing it.]

20. Relative values of the functions. From group (B) we derive the following:

that is, The tangent of any angle equals the ratio of the sine

of that angle to its cosine; and

The cotangent of an angle equals the ratio of the cosine to the sine.

Also from the same group we find the following combinations of the functions equal to unity:

21. Each function may be expressed in terms of any one of the other seven. Thus, from equations (3), (5), (8), of the preceding article we have directly

which gives the value of sin x in terms of cos, csc, and cvs. To find it in terms of the tangent, we have from (1),

and so on for the other two functions. In this way the following table may be formed.

(10)

ھی

EXERCISES.

0.567 ?

1. In what quadrants may A be if sin A
2. If sin x =
-, find the other functions when the terminal line is in
the third quadrant.

3. Find the functions when tan x = 2, and the terminal line is in the

third quadrant.

4. Find x when sin x cos x.

[Make cos x = √1 — sin2 x, solve and compare the result with exercise 3, page 8.]

5. Find x when sin x = tan x.

Ans. x=

0.

6. Find the trigonometrical functions corresponding to cot x = 2.
7. If tan x= 2 cos x, find sin x in terms of cot x.

8. If cot x 2 tan x, find sin x in terms of cos x.
9. Find sin x from the equation 2 sin2 x − 3 sin x =

= 1.

[Here sin x is the unknown quantity, and the equation is to be
solved as a complete quadratic.]

10. Find tan x from the equation a tan x + b cot x = c.
[Substitute cot x from equation (4) and solve.]

x) + cos2 (1 — дx) equal?

15. In a right triangle, if the base be 3a and the perpendicular 4a, find

each of the eight trigonometrical functions.

16. Given 2 sin x = = 3 cos y, and tan y = 3 cos x, to find sin x.

[We have from equation (1) tan y

sin y
which, by means of
cos y'

11- cos2 y

and this in the second of

Cos y

equation (3), becomes
the given equations gives

1 cos2 y = 9 cos2 x cos2 y,

from which finding cos y and substituting in the first of the given equations, gives sin x = ± 0.89 + or ± 0.56 +.]

17. Given sin x + cos y = 1, and sin x cos y

=

, find sin x and sin y. 18. Solve the simultaneous equations tan x cot y = 2, and sin x cos

y = 1.

19. Find sin x from the equations asin = bcosy, and sina x = cos y. [From the first equation, sin x log a = cos y log b; from the

8

20. Find y from the equations (tan x)coty = 5, (tan x)cosy = }. 21. Find the values of x from the equation

sin x cos x tan x=0.

[Each factor may be zero, hence sin x = 0, and x = 0, π, 2π, &c.
If cos x 0, then дx = π, π, &c., and x = π, зπ, 5π, &c.
If tan x = 0, then дx = 0, π, 2π, &c., and x = 0, 4ñ, 8π, &c.]

22. Functions of negative angles. If the angle be negative, and numerically less than 90°, it will be below. the line OX in the figure on page 21, and the ordinate o will be negative, oro, while the abscissa will be positive, or +a; and we have