How do you find the exact value of #sin45cos30-cos45sin30# using the sum and difference, double angle or half angle formulas?

2 Answers
Nghi N.
Aug 25, 2016

#sin 15 = (sqrt2/4)(sqrt3 - 1)#

Explanation:

Use trig identity:
sin (a - b) = sin a.cos b - sin b.cos a
Trig table -->
#sin 45 = sqrt2/2# and #cos 30 = sqrt3/2#
#sin 30 = 1/2#, and #cos 45 = sqrt2/2#
There for:
#sin (45 - 30) = sin 15 = (sqrt2/2)(sqrt3/2) - (sqrt2/2)(1/2) = #
#= sqrt2/4(sqrt3 - 1)#

P dilip_k
Aug 25, 2016

Using following identities to evalute

(1)#" "2sinAcosB=sin(A+B)+sin(A-B)#

(2)#" "2cosAsinB=sin(A+B)-sin(A-B)#

(3)#" "sinA=sqrt(1/2(1-cos2A))#

#sin45cos30-cos45sin30#

#=1/2(2sin45cos30-2cos45sin30)#

#=1/2(sin(45+30)+sin(45-30)- (sin(45+30)-sin(45-30)) #

#=1/2(sin75+sin15-sin75+sin15)#

#=1/2xx2sin15=sin15#

Or by usung directly the identity
#sinAcosB-cosAsinB=sin(A-B)#

we get the given expression

#=sin(45-30)=sin15#

#=sqrt(1/2(1-cos(2*15))#

#=1/sqrt2sqrt(1-sqrt3/2)#

#=1/sqrt2sqrt(1/4(4-2sqrt3))#

#=1/(2sqrt2)sqrt((sqrt3-1)^2)#

#=1/(2sqrt2)(sqrt3-1)#

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