As soon, then, as the sine, cosine, and tangent of an angle are known, their reciprocals the cosecant, secant, and cotangent may easily be obtained.

11. Problem. To find the tangent when the sine and cosine of an angle are known.

Solution. The quotient of sin. A divided by cos. A is, by equations (4),

Snm of squares of sine and cosine.

12. Corollary. Since the cotangent is the reciprocal of the tangent, we have

cotan. A =

cos. A

sin. A'

(8)

13. Problem. To find the cosine of an angle when its sine is known.

Solution. We have, by the Pythagorean proposition, in the right triangle ABC (fig. 4.)

that is, the sum of the squares of the sine and cosine

(12)

or

whence

(cosec. A)2(cotan. A)2 = 1;

(cosec. A)21+ (cotan. 4)2.

16. Scholium. The whole difficulty of calculating the trigonometric tables of sines and cosines, tangents and cotangents, secants and cosecants is, by the preceding propositions, reduced to that of calculating the sines alone.

17. EXAMPLES.

1. Given the sine of the angle A, equal to 0.4568, calculate its cosine, tangent, cotangent, secant, and cosecant.

Solution. By equation (10)

cos. A = √1— (sin. A)2 = √ ( 1 + sin. A) (1 — sin. A).

log. sec. A- log. cos. A = 0.05082,

sec. A = 1.1241,

log. cosec. A=— log. sin. A = 0.34027,

cosec. A = 2.1891.

2. Given sin. A = 0.1111; find the cosine, tangent, cotangent, secant, and cosecant of A.

3. Given sin. A = 0.9891; find the cosine, tangent,

cotangent, secant, and cosecant of A.

Sine, &c. in the circle whose radius is unity.

18. Theorem. The sine of an angle is equal to the perpendicular let fall from one extremity of the arc, which measures it in the circle, whose radius is unity, upon the radius passing through the other extremity. Proof. Let BCA (fig. 5.) be the angle, and let the radius of the circle AB A'A be

Let fall, on the radius AC, the perpendicular BP, and we have by $5, in the right triangle BCP,

19. Theorem. In the circle of which the radius is unity, the cosine of an angle is equal to the portion of the radius, which is drawn perpendicular to the sine, included between the sine and the centre.

Proof. For if BCA (fig. 5.) is the angle, we have, by $9,

20. Theorem. In the circle of which the radius is unity, the secant is equal to the length of the radius drawn through one extremity of the arc which measures the angle, and produced till it meets the tangent drawn through the other extremity.

The trigonometric tangent is equal to that portion of the tangent, drawn through one extremity of the arc, which is intercepted between the two radii which terminate the arc.