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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.
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."
INVERSE TRIGONOMETRIC FUNCTIONS
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
95. The expressions sin-ly, cos-1y, etc., are called the Inverse Trigonometric Functions or Inverse Circular Functions.
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,
tan ly, csc-ly, cot-ly, we shall understand to be an angle, either positive or negative, whose magnitude is not greater than 90°.
(2) When we are given the equations : = cos1y, sec-1y, we shall limit to a positive angle whose magnitude is not greater than 180°.
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
By our agreements we know that = 135°. We can now write
· ( — — — 2 )·
If, on the other hand, we are given
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.
EXERCISE. terms of x.
But if we had given us
sin 0 ;
= csc 0;
we conclude = 45°.
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
98. Let x = sin 0.
99. To express two inverse trigonometric functions as a single inverse function;
e.g. consider sin-1x + sin-1y.
.'. sin-1x =
.. sin (A + B) = x√1 − y2 + y√1 — x2.
A + B = sin-1x + sin-1y = sin-1 (x√1 − y2 + y√1 − x2).
Express tan-1x + tan-ly as a single inverse trigonometric
tan-1 (}); find cos p.
sin-1 (-1) find sec; find tan p.
0 = tan-1
2 m + 1
x = sin A; .. A sin-1x.
y sin B; .. B= sin-1y;
10. cos ̄1x
b + a
2. 4 = cos−1 ( − }) find csc p.
; show that (@+4)=—
cos-1y = cos-1 (xy ± √(1 − x2) (1 — y2)).
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.
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°.
102. Also if A be an angle of a triangle, then A may have any value between 0° and 180°. Hence,
(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.