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Since covers A=1. sin A, we have sin A=1

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We then take the opposite side BC equal to 3 units, and the hypotenuse AB equal to 5 units.

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6. vers A

EXAMPLES.

In each case find the values of the remaining functions:

3. tan A =

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16. To find the values of the sine, cosine, tangent, cotangent, secant, and cosecant of 45°.

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Let ABC be an isosceles right triangle, having each of the sides AC and BC equal to 1.

Then, AB = √AC2 + BC2 = √1 + 1 = √2.

Also, the angles A and B are equal; and since their sum is a right angle, each is equal to 45°.

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The second line might have been derived from the first by aid of Art. 14, since 45° is complement of itself.

17. To find the values of the sine, cosine, tangent, cotangent, secant, and cosecant of 30° and 60°.

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Let ABD be an equilateral triangle, having each side equal to 2. Draw AC perpendicular to BD; then by Geometry,

BC= 1BD = 1, and ≤ BAC = }≤ BAD=30°.

AB2

Again, AC-VAB - BC2 = √4-1 = √/3.

Then from the triangle ABC, by definition,

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Or the functions of 60° may be derived from those of 30° by aid of Art. 14, since 60° is the complement of 30°.

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18. We obtain from the definitions of Art. 10,

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That is, the sine of an acute angle is the reciprocal of the cosecant, the tangent is the reciprocal of the cotangent, and the secant is the reciprocal of the cosine.

19. To prove the formula

sin2 A+ cos2 A= 1.

Note. Sin2 A signifies (sin A)2; that is, the square of the sine of A

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Whence by definition,

or,

(sin A)+(cos A)2 = 1;

sin2 A+ cos2 A= 1.

(4)

The result may be written in the forms

sin A= √1 - cos2 A, and cos A= √1 - sin2 A.

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21. To prove the formula

sec2 4=1+tan2 A, and csc2 A=1+ cc t2 A.

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III. APPLICATION OF ALGEBRAIC SIGNS. .

TRIGONOMETRIC FUNCTIONS OF ANGLES
IN GENERAL.

22. In Geometry we are, as a rule, concerned with angles. less than two right angles; but in Trigonometry it is convenient to regard them as unrestricted in magnitude.

Α'

B

A

B'

A"

In the circle AA", let AA" and A'A"" be a pair of perpendicular diameters.

Suppose a radius OB to start from the position OA, and revolve about the point O as a pivot in the direction of OA'. When it coincides with OA', it has described an angular magnitude of 90°; when it coincides with OA", of 180°; with OA"", of 270°; with OA, its starting-point, of 360°; with OA' again, of 450°; etc. We thus see that a significance may be attached to any positive angular magnitude.

23. The interpretation of an angle as the measure of the amount of rotation of a moving radius, enables us to distinguish between positive and negative angles.

Thus, if a positive angle is taken to indicate revolution from the position OA in the direction of OA', a negative angle would naturally be taken as signifying revolution from the position OA' in the opposite direction, or towards OA"".

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