For x,y,z,t∈R,$$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$y+$$$$sec$$−$$1$$⁡z≥$$t2$$−$$2$$$$π$$$$t+3$$$$π$$The value of x $$+$$ y $$+$$ z is equal toThe principal value of $$cos$$−$$1$$⁡$$($$$$cos$$⁡$$5t2)$$ isThe value of $$cos$$−$$1$$⁡(min{x$$,y,$$z}) is

Question

For x,y,z,t∈R,$$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$y+$$$$sec$$−$$1$$⁡z≥$$t2$$−$$2$$$$π$$$$t+3$$$$π$$The value of x $$+$$ y $$+$$ z is equal toThe principal value of   $$cos$$−$$1$$⁡$$($$$$cos$$⁡$$5t2)$$ isThe value of $$cos$$−$$1$$⁡(min{x$$,y,$$z}) is

Infinity Learn · Accepted Answer

$$sin$$−$$1$$⁡x∈[−π$$2$$,π$$2$$]$$cos$$−$$1$$⁡y∈[$$0$$,π] $$sec$$−$$1$$⁡z∈[$$0$$,π$$2)$$∪$$($$π$$2$$,π]⇒ $$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$y+$$$$sec$$−$$1$$⁡z≤π$$2+$$π$$+$$π$$=5$$$$π$$$$2Also$$,$$t2$$−$$2$$$$π$$$$t+3$$π$$=t2$$−$$2$$$$π$$$$2t+$$π$$2$$−π$$2+3$$π $$=(t$$−π$$2)2+5$$$$π$$$$2$$≥$$5$$$$π$$$$2The$$ given inequation exists if equality holds, i.e.,  L.H.S.$$=R$$.H.S.$$=5$$$$π$$$$2$$⇒$$x=1$$,$$y=$$−$$1$$,$$z=$$−$$1andt=$$π$$2$$    $$sin$$−$$1$$⁡x∈[−π$$2$$,π$$2$$]$$cos$$−$$1$$⁡y∈[$$0$$,π] $$sec$$−$$1$$⁡z∈[$$0$$,π$$2)$$∪$$($$π$$2$$,π]⇒ $$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$y+$$$$sec$$−$$1$$⁡z≤π$$2+$$π$$+$$π$$=5$$$$π$$$$2Also$$,$$t2$$−$$2$$$$π$$$$t+3$$π$$=t2$$−$$2$$$$π$$$$2t+$$π$$2$$−π$$2+3$$π $$=(t$$−π$$2)2+5$$$$π$$$$2$$≥$$5$$$$π$$$$2The$$ given inequation exists if equality holds, i.e.,  L.H.S.$$=R$$.H.S.$$=5$$$$π$$$$2$$⇒$$x=1$$,$$y=$$−$$1$$,$$z=$$−$$1andt=$$π$$2$$⇒$$cos$$−$$1$$⁡$$($$$$cos$$⁡$$5t2)=$$$$cos$$−$$1$$⁡$$($$$$cos$$⁡(5$$$$π$$$$2))=$$π$$2$$  $$sin$$−$$1$$⁡x∈[−π$$2$$,π$$2$$]$$cos$$−$$1$$⁡y∈[$$0$$,π] $$sec$$−$$1$$⁡z∈[$$0$$,π$$2)∪$$($$π$$2$$,π]⇒ $$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$y+$$$$sec$$−$$1$$⁡z≤π$$2+$$π$$+$$π$$=5$$$$π$$$$2Also$$,$$t2$$−$$2$$$$π$$$$t+3$$π$$=t2$$−$$2$$$$π$$$$2t+$$π$$2$$−π$$2+3$$π $$=(t$$−π$$2)2+5$$$$π$$$$2$$≥$$5$$$$π$$$$2The$$ given inequation exists if equality holds, i.e.,  L.H.S.$$=R$$.H.S.$$=5$$$$π$$$$2$$⇒$$x=1$$,$$y=$$−$$1$$,$$z=$$−$$1andt=$$π$$2$$⇒$$cos$$−$$1$$⁡$$($$$$cos$$⁡$$5t2)=$$$$cos$$−$$1$$⁡$$($$$$cos$$⁡$$(5$$$$π$$$$2))=$$π$$2$$ $$cos$$−$$1$$⁡$$(min{x$$,y,$$z})=$$$$cos$$−$$1$$⁡$$($$−$$1)=$$π

For x,y,z,t∈R,sin−1⁡x+cos−1⁡y+sec−1⁡z≥t2−2πt+3πThe value of x + y + z is equal toThe principal value of cos−1⁡(cos⁡5t2) isThe value of cos−1⁡(min{x,y,z}) is

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