Home > Trigonometry > The Law of Cosines

If 3costheta+4sintheta=Asin(theta+alpha),then values of A and alpha are?

Answer 1

Zoe Austin

#A=5# and #alpha=tan^(-1)(3/4)#

#3costheta+4sintheta#

= #4sintheta+3costheta#

= #5(sinthetaxx4/5+costhetaxx3/5)#

= #5(sinthetacosalpha+costhetasinalpha)#, where #alpha=tan^(-1)(3/4)#

= #5sin(theta+alpha)#

Hence #A=5# and #alpha=tan^(-1)(3/4)#

Note: #-# When we have #asintheta+bcostheta#, we take #sqrt(a^2+b^2)# outside and then weare left with #a/sqrt(a^2+b^2)# as #cosalpha# and #b/sqrt(a^2+b^2)# as #sinalpha#. Note that then #sin^2alpha+cos^2alpha=1# and #tanalpha=b/a#. Here we have #a=4# and #b=3# and hence we have #sqrt(3^2+4^2)=5# outside and #cosalpha=4/5# and #sinalpha=3/5# and #tanalpha=3/4# and hence #alpha=tan^(-1)(3/4)#.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.