Let $$M=sin4$$θ−$$1$$−$$sin2$$θ$$1+cos2$$θ$$cos4$$θ$$=$$α$$I+$$βM−$$1$$ , Where α$$=$$αθ and β$$=$$βθ are real numbers, and I is the $$2$$×$$2$$ identity matrix. If α$$*$$ is the minimum of the set αθ:θ∈[$$0$$,$$2$$π$$)$$ and β$$*$$ is the minimum of the set βθ:θ∈[$$0$$,$$2$$π$$)$$. Then the value of α$$*+$$β$$*$$ is

Question

Let $$M=sin4$$θ−$$1$$−$$sin2$$θ$$1+cos2$$θ$$cos4$$θ$$=$$α$$I+$$βM−$$1$$ , Where α$$=$$αθ  and β$$=$$βθ  are real numbers, and I is the $$2$$×$$2$$ identity matrix. If α$$*$$  is the minimum of the set αθ:θ∈[$$0$$,$$2$$π$$)$$  and β$$*$$   is the minimum of the set βθ:θ∈[$$0$$,$$2$$π$$)$$. Then the value of α$$*+$$β$$*$$  is

Infinity Learn · Accepted Answer

$$M=sin4$$θ$$-1-sin2$$θ$$1+cos2$$θ$$cos4$$θ$$=$$α$$I+$$β$$M-1$$ Where α$$=$$α$$($$θ$$)$$ and β$$=$$β$$($$θ$$)$$ are real numbers |M|$$=($$$$sin$$θ$$cos$$θ$$)4+1+sin2$$θ$$1+cos2$$θ$$=2+sin2$$θ$$cos2$$θ$$+sin4$$θ$$cos4$$θ since $$M=$$α$$I+$$β$$M-1sin4$$θ$$-1+sin2$$θ$$1+cos2$$θ$$cos4$$θ$$=$$α$$00$$α$$+$$β|M|$$cos4$$θ$$1+sin2$$θ$$-1-cos2$$θ$$sin4$$θ Comparing on both side $$sin4$$θ$$=$$α$$+$$β|M|$$cos4$$θ$$-1-sin2$$θ$$=$$β|M|$$1+sin2$$θ⇒β$$=-$$|M| $$=-2+sin2$$θ$$cos2$$θ$$+sin4$$θ$$cos4$$θ  therefore $$sin4$$θ$$=$$α$$-cos4$$θ ⇒α$$=$$αθ$$=sin4$$θ$$+cos4$$θ$$=1-2sin2$$θ$$cos2$$θ$$=1-12sin22$$θα$$*=$$ minimum of α$$($$θ$$)=1-121=12now$$ β$$*$$ $$=Minimum$$ of βθ$$=-2+sin2$$θ$$cos2$$θ$$+sin4$$θ$$cos4$$θ$$=-sin2$$θ$$cos2$$θ$$+122+74=-14sin22$$θ$$+122+74$$β$$*=-14+122+74=-3716$$  when θ$$=n$$π$$+$$π$$4$$α$$*+$$β$$*=-3716+12=-37+816=-2916$$

Let M=sin4θ−1−sin2θ1+cos2θcos4θ=αI+βM−1 , Where α=αθ and β=βθ are real numbers, and I is the 2×2 identity matrix. If α* is the minimum of the set αθ:θ∈[0,2π) and β* is the minimum of the set βθ:θ∈[0,2π). Then the value of α+β is

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Let M=sin4θ−1−sin2θ1+cos2θcos4θ=αI+βM−1 , Where α=αθ and β=βθ are real numbers, and I is the 2×2 identity matrix. If α* is the minimum of the set αθ:θ∈[0,2π) and β* is the minimum of the set βθ:θ∈[0,2π). Then the value of α*+β* is