How many common points have #f# and #g# function graphs in the interval #<0;2pi># ?

a)
#f(x)=cosx#
#g(x)=1-sinx#

I can do this by graphing, but I want to know how to do this without graphs.

1 Answer
Noah G
Oct 22, 2017

intersection points are #(0, 1)# and #(pi/2, 0)#

Explanation:

We can solve algebraically. We know that #y = cosx# and #y = 1- sinx#, so we can substitute #f(x)# into #g(x)# or vice versa.

#cosx =1 - sinx#

#cosx + sinx = 1#

#cos^2x + sin^2x + 2sinxcosx = 1#

#1 + 2sinxcosx = 1#

#2sinxcosx = 0#

#sin(2x) = 0#

#2x = 0 or pi#

#x = 0 or pi/2#

We check to make sure neither roots are extraneous:

#cos(0) + sin(0) =^? 1#

#1 + 0 = 1 color(green)(√)#

ALSO

#cos(pi/2) + sin(pi/2) =^? 1#

#0 + 1 = 1 color(green)(√)#

Hence, the graphs intersect at #(0, 1)# and #(pi/2, 0)#. The graph confirms.

enter image source here

Hopefully this helps!

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