Find the exact value of cos(alpha minus beta) if sin alpha equals 0.1102 and cos beta equals 0.4111, and both alpha and beta are first-quadrant angles?

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Answer 1 · 2018-03-01T05:52:28

#color(red)(cos(alpha-beta)=0.5091)#

.

#sinalpha=0.1102#

#cosbeta=0.4111#

Using #sin^2alpha+cos^2alpha=1#, we get:

#cos^2alpha=1-sin^2alpha=1-(0.1102)^2=1-0.01214404=0.98785596#

#cosalpha=sqrt0.98785596=+-0.9939#

We only use the positive value because the angle is in quadrant I:

#cosalpha=0.9939#

#sin^2beta=1-cos^2beta=1-(0.4111)^2=1-0.1690021=0.83099679#

#sinbeta=sqrt0.83099679=+-0.9116#

We only use the positive value because the angle is in quadrant I:

#sinbeta=0.9116#

#cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta#

#cos(alpha-beta)=(0.9939)(0.4111)+(0.1102)(0.9116)=0.4086+0.1005=0.5091#

Find the exact value of cos​(alpha minus beta​) if sin alpha equals 0.1102 and cos beta equals 0.4111​, and both alpha and beta are​ first-quadrant angles?