The secant and cosecant are positive if measured in the same direction as the terminal line, OP; negative if measured in the opposite direction.

The signs of the functions of angles in the different quadrants are as follows:

9. It is evident that the values of the functions of an angle depend only upon the position of the sides of the angle. If two angles differ by 360°, or any multiple of 360°, the position of the sides is the same, hence the values of the functions are the same.

Thus in Fig. I the angle is 120°, in Fig. 2 the angle is 840°, yet the lines which represent the functions are the same for both angles.

EXERCISE

Determine, by drawing the necessary figures, the sign of tan 1000°; cos 810; sin 760°; cot -70°; cos - 550°; tan -560°; sec 300°; cot 1560°; sin 130°; cos 260°; tan 310°.

or

RELATIONS OF THE FUNCTIONS

10. By § 5, whatever may be the length of OP, we have

cot x=

cot x

tan x

The angle has been taken in the first quadrant; the results are, however, true for any angle. The proof is the same for angles in other quadrants, except that SP becomes negative in the third and fourth quadrants, and OS in the second and third.

(8.) Prove tan r sin x+cosx = secx.

12. The formulas (1)-(8) of § 10 are algebraic equations connecting the different functions of the same angle. If the value of one of the functions of an angle is given, we can substitute this value in one of the equations and solve to find another of the functions. Repeating the process, we find a third function, etc.

In solving equation (6), (7), or (8) a square root is extracted; unless something is given which determines whether to choose the positive or negative square root, we get two values for some of the functions. The reason for this is that there are two angles less than 360° for which a function has a given value.

EXERCISES

13. (1.) Given x less than 90° and sin x=1; find all the other functions of x.

Solution.

I

Since x is less than 90°, we know that cosx is positive.

; find the other functions of

- 30°.

=

(3.) Given sin (— 30°) = (4.) Given x in quadrant III and sinx=— ; find all the other functions of x.

(5.) Given y in quadrant IV and siny=-3; find all the other functions of y.

(6.) Given cos 60°; find all the other functions of 60°.

(7.) Given sin o°o; find cos o° and tan oo.

(8.) Given tanz=} and ≈ in quadrant I; find the other functions

of z.

(9.) Given cot 45°=1; find all the other functions of 45o.

(10.) Given tany=√5 and cosy negative; find all the other functions of y.

(11.) Given cot 30° = √3; find the other functions of 30o.

(12.) Given 2 sin x=1-cos x and x in quadrant II; find sin x and cosx.

(13.) Given tan x + cotx=3 and x in quadrant I; find sinx.

FUNCTIONS OF AN ACUTE ANGLE OF A RIGHT TRIANGLE

14. The functions of an acute angle of a right triangle can be expressed as ratios of the sides of the triangle.

Remark.-Triangles are usually lettered, as in Fig. 2, the capital letters denoting the angles, the corresponding small letters the sides opposite.

In the right triangle ABC, by $ 5,

sin A = = = cos B;

BC a

AB с

15. From § 14, for an acute angle of a right triangle, we have