How do you use the angle sum or difference identity to find the exact value of #csc(pi/3+pi/4)#?

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Answer 1 · 2017-02-01T02:02:42

# csc(pi/3+pi/4) = 4/(sqrt(2)(1+sqrt(3))) #

Using the summation formula:

# sin(A+B)=sinAcosB+cosAsinB#

We have:

# sin(pi/3+pi/4)= sin(pi/3)cos(pi/4)+cos(pi/3)sin(pi/4)#
# " "= sqrt(3)/2sqrt(2)/2+1/2sqrt(2)/2#
# " "= (sqrt(2)(1+sqrt(3)))/4#

Then;

# csc(pi/3+pi/4) = 1/sin(pi/3+pi/4) #
# " "= 4/(sqrt(2)(1+sqrt(3))) #

Which can easily be confirmed with a calculator.