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94. From the equation y sin o, we know that is an angle whose sine is y. The last statement is expressed by the notation + sin 1y.

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Hence sin-ly is an angle.

sin-ly is sometimes read, "an angle whose sine is y," sometimes "arc-sine y," but more frequently "anti-sine y." But it must be remembered that it means "an angle whose sine is y."

CHAPTER XII

INVERSE TRIGONOMETRIC FUNCTIONS

99

cos-ly means "an angle whose cosine is y." tan-ly means "an angle whose tangent is y. csc-ly means "an angle whose cosecant is y." sec-ly means "an angle whose secant is y."

cot-ly means "an angle whose cotangent is y."

These are read "anti-cosine y," "anti-tangent y," "anti-cosecant y," "anti-secant y," "anti-cotangent y," respectively.

EXAMPLE. 30° = sin

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45° =

95. The expressions sin-ly, cos-1y, etc., are called the Inverse Trigonometric Functions or Inverse Circular Functions.

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96. In Art. 58 we showed that an infinite number of angles, differing by 2, have the same ratios. Accordingly an infinite number of angles will satisfy an equation of the form = sin-1y, 0 = cos-1x, etc. Accordingly for the sake of definiteness we shall (unless otherwise stated) make the following conventions:

(1) When we are given either of the equations sin-'y,

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tan ly, csc-ly, cot-ly, we shall understand to be an angle, either positive or negative, whose magnitude is not greater than 90°.

1

√2

= sin-1.

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(2) When we are given the equations : = cos1y, sec-1y, we shall limit to a positive angle whose magnitude is not greater than 180°.

68

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With these agreements one value and but one will satisfy any of these equations.

If, however, is given, we can always write a definite equation.

For example, 225° = sin-1

and had known nothing else whatever of ø, by our agreements we would have concluded = − 45°.

EXAMPLE 2. Given : = cos-1

8-11-20)

By our agreements we know that = 135°. We can now write

· ( — — — 2 )·

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If, on the other hand, we are given

= sin-1

97. To express the inverse ratios in terms of a given one. e.g. if o tan-', to express the inverse trigonometric functions in terms of x.

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EXERCISE. terms of x.

135° sin-1

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But if we had given us

sin 0 ;

= csc 0;

we conclude = 45°.

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No ambiguity of sign exists.

For if x is positive, sin is positive; therefore √1+22 is positive. If x is negative, sin is negative; hence V1+ is positive. Then by convention cos @ is positive; therefore √1+ is positive.

If x = cos 0, express the inverse trigonometric functions in

Then

98. Let x = sin 0.

EXERCISE. Show

Put

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99. To express two inverse trigonometric functions as a single inverse function;

e.g. consider sin-1x + sin-1y.

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.'. sin-1x =

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π

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2

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.. sin (A + B) = x√1 − y2 + y√1 — x2.

A + B = sin-1x + sin-1y = sin-1 (x√1 − y2 + y√1 − x2).

EXERCISE.

tan-1x

X sec-1x

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Express tan-1x + tan-ly as a single inverse trigonometric

tan-1 (}); find cos p.

sin-1 (-1) find sec; find tan p.

1

0 = tan-1

2 m + 1

x = sin A; .. A sin-1x.

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π

= cot-1x;

2

y sin B; .. B= sin-1y;

=

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π

2

10. cos ̄1x
11. tan-1 tan-1 + tan-13.

cos-x.

csc-1x.

EXAMPLES. XXV.

b + a
- x
26

2. 4 = cos−1 ( − }) find csc p.

; show that (@+4)=—

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cos-1y = cos-1 (xy ± √(1 − x2) (1 — y2)).
12. tan-12+ tan

tan-11=2

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CHAPTER XIII

ON THE RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE

100. The three sides and the three angles of any triangle are called its six parts.

By the letters A, B, C we shall indicate

geometrically, the three angular points of the triangle ABC; algebraically, the three angles at those angular points respectively.

A

B

a
FIG. 49.

By the letters a, b, c, we shall indicate the measures of the sides BC, CA, AB, opposite the angles A, B, C, respectively.

101. I. We know that A + B + C = 180°.

(Geom.)

102. Also if A be an angle of a triangle, then A may have any value between 0° and 180°. Hence,

1

(i.) sin ▲ must be positive (and less than 1);

(ii) cos ▲ may be positive or negative (but must be numerically less than 1);

(iii.) tan A may have any value whatever, positive or negative.

103. Also, if we are given the value of

(i.) sin A, there are two angles, each less than 180°, which have the given positive value for their sine.

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