development of trig identities? easy 10 points!?

Question

alright, i understand that by drawing 2 right angle triangles in the first and second quadrants, with the sides labelled (a), (b), and (root of all a^2 + b^2) i can prove that
sin (pi / 2 + theta) = cos theta...........but now
using the same method, i need to prove that
csc (180 degrees + theta) = - csc theta

i dont understand how this is proven because everytime i draw my diagram, it doesnt work...


or how do i prove that
sin (pi - theta) = sin theta????

thanks in advance... i missed 2 days of classes and im so confused!

Answer 1

Question 1
cosec ( 180 + θ )
1 / sin (180 + θ )
1 / (sin180° cos θ + cos180° sin θ )
1 / (- sin θ )
- cosec θ

Question 2
sin ( π - θ )
sin π cos θ - cos π sin θ
0 - (-1) sin θ
sin θ

Answer 2

use the certainty that csc =a million/sin = r/y and cot = a million/tan=x/y [ (a million + cot^2 theta) / (csc^2theta) ] = a million [ (a million+(x/y)^2 ) / (r/y)^2 ] = a million Separate into 2 fractions [ a million / (r/y)^2 ] + [ (x/y)^2 / (r/y)^2 ] =a million [ a million / (r/y)^2 ] + [ (x^2/y^2) * (y^2/r^2) ] =a million [ (y/r)^2 ] + [ (x/r)^2] = a million look elementary? replace y/r and x/r with sin and cos sin^2 theta + cos^2 theta = a million it extremely is an identity. a million=a million 2. i did no longer particularly get to an answer yet i'm going to instruct what i all started and with any luck it will help somebody else. [ (csc theta) / (a million + csc theta)] + [ (csc theta) / (a million - csc theta)] = [(2 sin theta) / cos^2 theta)] [ (r/y) / (a million+(r/y)) ] + [ (r/y)/(a million-(r/y)) ] = [ (2y/r) / (x/r)^2 ] enable's simplify the LS first upload the fractions on the left by utilising looking the liquid crystal demonstrate - it extremely is (a million-(r/y)^2 ) [ (r/y)*(a million-(r/y)) + (r/y)*(a million+(r/y)) ] / (a million-(r/y)^2) [ r/y - (r/y)^2 + (r/y) + (r/y)^2 ] / (a million-(r/y)^2 ) 2(r/y) / (a million-(r/y)^2)

Answer 3

sin(a-b) =sina cosb -sinb cosa

sin(π -θ) =sinπ cosθ - sinθ cosπ
=0 * cos θ - (-1)* sinθ = sinθ

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to draw angles , use Cartesian coordinate system, where x represent cos & y represent sin