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Tignometry

1) Using logarithms find the area of the following triangles :
i) a = 12.7, b = 21.5,and c = 28.6
ii) c = 426, A= 45° 48' 36", and B = 61° 2' 13"
iii) An isosceles triangle in which each of the equal sides is 14.72 in. and the vertex angle 47° 28' .

2) Find the radius of the inscribed circle and the radius of the circumscribed circle for the triangle in part (i) of question (1)

3) Using exact values, without referring to the tables, find

a) tan 105° and, b) sin 255°

4) Find cos(x+y) when both X and Y are acute angles, and sin x= 3/5 and
sin y =5/13, without using tables.

5) Verify the following identities :

a) sin 2x = ( 2tan x / ( 1+ tan ^2 x

b) (sin 2 t - cos 2 t / cos t ) = sec t , where ' t ' stands for theta.

Solution Preview

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1) Using logarithms find the area of the following triangles :
i) a = 12.7, b = 21.5,and c = 28.6
ii) c = 426, A= 45° 48' 36", and B = 61° 2' 13"
iii) An isosceles triangle in which each of the equal sides is 14.72 in. and the vertex angle 47° 28' .

(i) s = (a + b + c)/2 = (12.7 + 21.5 + 28.6)/2 = 31.4
Area = √[s(s - a)(s - b)(s - c)] = √[31.4 * (31.4 - 12.7) * (31.4 - 21.5) * (31.4 - 28.6)]
Area = √(31.4 * 18.7 * 9.9 * 2.8)
log(Area) = log [√(31.4 * 18.7 * 9.9 * 2.8)]
log(Area) = log (31.4 * 18.7 * 9.9 * 2.8)^(1/2) = (1/2) log(31.4 * 18.7 * 9.9 * 2.8)]
= (1/2) [log 31.4 + log 18.7 + log 9.9 + log 2.8] = (1/2) [1.4969 + 1.2718 + 0.9956 + 0.4472] = 2.1058
∴ Area ...

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