<prompt> Acha mathbfD iwe matrix inayowakilisha upanuzi wenye kipimo cha k 0, na mathbfR iwe matrix inayowakilisha mzunguko kuhusu asili kwa pembe ya theta counter -saa. KamamathbfR mathbfD anzapmatrix 8 -4 4 8 mwishopmatrix,kisha tafuta tan theta.  <response>: Tunayo mathbfD anzapmatrix k 0 0 k endpmatrix na mathbfR anzapmatrix cos theta -sin theta sin theta cos theta mwishopmatrix, hivyomathbfR mathbfD anzapmatrix cos theta -sin theta dhambi theta cos theta mwishopmatrix anzapmatrix k 0 0 k mwishopmatrix anzapmatrix k cos theta -k sin theta k sin theta k cos theta mwishopmatrix. Kwa hivyo, k cos theta 8 na k sin theta 4. Tukigawanya milinganyo hii, tunapata tan theta frac12. 