4 Express the circumfcrence of tsc cirele as a function of t

4. Express the circumfcrence of tsc cirele as a function of the angle I Note: The hypotenuse of the right triangle is the diameter of the eircle-] 5. Express the length of (a+b) in terms of e 6FT 6. Express the area of an isosceles right triangle in terms of its hypotenuse. A sector ofa circle has central angle (a) Express the arclength a, in terms of r 7. and the area ofthe sector ofa circle formed by the angle as a function of r and .

Solution

looking at the right angled triangle

hypotenuse = diameter

applying cos rule on the triangle

cos theta = base / hypotenuse

base = 12

cos theta = 12 / d

d = 12 / cos theta

circumference = pi*d

circumference in terms of theta = pi * 12 / cos theta

5) cos theta = 9/a

a = 9 / cos theta

then applying sine rule on 2nd triangle

since the two angles are corresponding

so sin theta = 6/b

b = 6 / sin theta

hence , a+b = ( 9/ cos theta + 6 / sin theta )

6) area of isoseles triangle is given by

1/ * base * height

hypotenuse = sqrt ( height^2 + base / 2^2)

hypotenuse^2 = height^2 + base^2 / 4

area = ( 1/ 4) * h^2

where h is the hypotenuse