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58. Find the Sine and Cosine of 36o.

cos 36°=1- 2 sin’18°

[(3) of Art. 49] 6 – 2 V5_15+1 =1 8

4

... sin 36°= V1- cos36°

V10 — 275

4

The above results are also the sine and cosine, respectively, of 54° (Art. 15).

Otherwise thus : Let x = 36°; then 2x=72°, and 3x=108°.

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But 36° is an acute angle, and therefore its cosine is positive.

V5+1. ... cos 36° =

4

59. If A+B+C = 180°, or if A, B, C are the Angles of a Triangle, prove the Following Identities :

A B C. (1) sin A + sin B + sin C

= 4cos

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COS

COS

A B В.
(2) cos A + cos B + cos C=1+4 sin sin sin

2

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(3) tan A + tan B + tan C:

= tan A tan B tan C.

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= 2cos. C

С C and

sin C=2 sin
COS

(Art. 49)
2
A+B С
COS

(Art. 15) 2 С A-B

С

A+B .. sin A + sin B + sin C: = 2 cos

COS

+2 cos COS 2 2

2

= 2 cos

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A+B A B Again, cos A + cos B= 2 cos COS

2

2

=2sins

A - B
COS

2

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Again, tan (A + B)=-tan C

(Art. 30)

tan A + tan B 1-tan A tan B

(Art. 47)

.. tan A + tan B => – tan C (1 – tan A tan B).

... tan A+tanB+tanc = tan A tan B tan C

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Note. — The student will observe that (1), (2), and (3) follow directly from Examples 1 and 2, and formula (3), respectively, of Art. 48, by putting

A +B+C = 180o.

EXAMPLES.

Prove the following statements if A + B + C = 180°: 1. cos (A + B – C)=- cos 2 C.

A B С 2. sin A + sin B – sinC=4sin sin

2 2 2

COS

3. sin 2 A +sin 2B + sin 2C=4sin A sin B sin C.

4. sin 2 A + sin 2 B – sin 2C=4sin C cos A cos B.

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60. Inverse Trigonometric Functions. — The equation sin A = x means that 0 is the angle whose sine is x; this may be written 0 = sin-?x, where sin-?x is an abbreviation for the angle (or arc) whose sine is x.

So the symbols cos-?x, tan-'x, and sec-ly, are read “the angle (or arc) whose cosine is 2," “ the angle (or arc) whose tangent is x," and "the angle (or arc) whose secant is y." These angles are spoken of as being the inverse sine of X, the inverse cosine of x, the inverse tangent of x, and the inverse secant of y, respectively. Such expressions are called inverse trigonometric functions.

NOTE. The student must be careful to notice that – 1 is not an exponent, sin-2 x

1 is not (sin x)"), which

sin x Notice also that sin-1V3.

is not an identity, but is true only for the par2

2 ticular angle 600.

This notation is only analoyous to the use of exponents in multiplication, where we have a-la= a - =1. Thus, cos-l (cos x) = x, and sin (sin-1 x) = x; that is, cos-1 is inverse to cos, and applied to it annuls it; and so for other functions.

= cog-11

The French method of writing inverse functions is arc sinx, arc cos x, arc tan x, and so on.

EXAMPLES.

1. Show that 30° is one value of sin-4.

We know that sin 30°= 1. .. 30° is an angle whose sine is }; or 30°= sin-'1.

2. Prove that tan-?? + tan-}=45°.

tan-} is one of the angles whose tangent is t, and tan-' } is one of the angles whose tangent is.

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Therefore 45° is one value of tan-'1 + tan-' .

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Any relations which have been established among the trigonometric functions may be expressed by means of the inverse notation. Thus, we know that

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5. By Art. 49, cos 20=2 cos0 – 1, which may be written 20=cos-'(2 cos? 0 – 1).

(— Put cos 0 = x. .: 2 cos- x = cos(2x - 1).

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