In the figure below, line AB is tanget to the circle with center E at point D. If the length of minor arc DC is 7π and line EC=21, what is the measure of angle BDC?

Question

anonymous · Answer

attachment

anonymous · Answer

ok so do you know what is the angle EDB ?

anonymous · Answer

no

anonymous · Answer

it is 90 degrees .. angle between tangent and radius is 90 degrees

anonymous · Answer

do you think that now you can solve it maybe ?

anonymous · Answer

does involve any trig?

anonymous · Answer

i think so

anonymous · Answer

if so, i'm new to to trig

anonymous · Answer

ok what kind is the triangle DCE ?

anonymous · Answer

equilateral

anonymous · Answer

are u able to read the diagram i attached here? i have pretty bad handwriting

anonymous · Answer

so it means that angle EDC = ECD

anonymous · Answer

and it means that DE = EC

anonymous · Answer

do you know the law of cosines ?

anonymous · Answer

c^=a^2b^2-2ab cos(C)

anonymous · Answer

good .. now we want to find EDC
sine if we subtract it from 90 we will get the requested angle..
so use this law for this angle

anonymous · Answer

c^2= 7π^2-21^2-2 (7π)(21) cos (90º) ?

anonymous · Answer

c = 21 (we got it from the fact that it is equilateral)

anonymous · Answer

and notice its not 90 degrees..

anonymous · Answer

the whole angle EDB = 90

anonymous · Answer

but EDC is not 90 .. call it C and solve the equation for cos(C)

anonymous · Answer

we're looking for angle BDC

anonymous · Answer

right .. but if we will have EDC 
we will be able to subtract it from 90 and get BDC

anonymous · Answer

since BDC + EDC = EDB = 90

anonymous · Answer

so how do we solve this equation?

anonymous · Answer

21^2= 7π^2+21^2-2 (7π)(21) cos (C)  right ?

anonymous · Answer

tell me if you agree with it

anonymous · Answer

ok. my previous equation applying the cosines law was wrong. and yes, i agree :)

anonymous · Answer

good .. so now isolate cos(C) and get the angle C

anonymous · Answer

solve for cos(C)?

anonymous · Answer

yes and eventually solve for C (by using arccos )

anonymous · Answer

C = π/42?

anonymous · Answer

mm i dont think so :
cos(C) = 7pi / 42
so
C is about 58.43

anonymous · Answer

so the answer is like 60 degrees?

anonymous · Answer

for C
.. but we need 90 - C  remember ?
so the answer for the question is about 30 or to be more accurate 31.57

anonymous · Answer

the closest is 30, bro?

anonymous · Answer

yes

anonymous · Answer

do you understand why we subtract C (60) from 90 and got 30 ?

anonymous · Answer

thnx, bro!

anonymous · Answer

yw :)

anonymous · Answer

ur reward is a medal