Identitas Trigonometri
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SOAL DAN PEMBAHASAN IDENTITAS TRIGONOMETRI By. Dafid Kurniawan
1. Sin A1-Sin2A+Cos A1-Cos2A=Sin3A+Cos3ASin2A.Cos2A Ruas Kiri =Sin A1-Sin2A+Cos A1-Cos2A=Sin ACos2A+Cos ASin2A=Sin A.Sin2ACos2A.Sin2A+Cos A.Cos2ASin2A.Cos2A=Sin3A+Cos3ASin2A.Cos2A=Ruas Kanan 2. 1-2Cos2ASin A.Cos A=Tan A-Cot A Ruas Kiri =1-2Cos2ASin A.Cos A=Sin2A+Cos2A-2Cos2A)Sin A.Cos A=Sin2ACos2ASin A.Cos A=Sin ACos A-Cos ASin A=Tan A-Cot A=Ruas Kanan 3.1+Sec ATan A-Tan ASec A=1+Sec ASec A.Tan A Ruas Kiri = 1+Sec ATan A-Tan ASec A=Sec A+Sec2A-Tan2ATan A.Sec A=Sec A+1+Tan2A-Tan2ATan A.Sec A=1+Sec ASec A.Tan A=Ruas Kanan Tan A.Cot A+Tan A=1+Tan2A= Sec2A 4. Tan A-Sin ASin3A=Sec A1+Cos A Ruas Kiri =Tan A-Sin ASin3A=Tan A-Sin ASin A. Sin2A=Sin ACosA-Sin ASin A. Sin2A=1Cos A.Sin2A-1Sin2A=1-Cos ACos A.Sin2A=1-Cos ACos A.(1-Cos2A)=1Cos ACos A.(1-Cos A)(1+Cos A)=1Cos A.(1+Cos A)=1Cos A.1(1+Cos A)=Sec A1+Cos A=Ruas Kanan 5. Sin3A+Cos2ACosec A=Sin A Ruas Kiri = Sin3A+Cos2ACosec A=Sin3A+Sin A.Cos2A=Sin ASin2A+Cos2A=Sin A=Ruas Kanan 6.(Sin A-Cos A)2+(Sin A+Cos A)2=2 Ruas Kiri =(Sin A-Cos A)2+(Sin A+Cos A)2=Sin2A-2Sin A.Cos A+Cos2A+Sin2A+2Sin A.Cos A+Cos2A=2Sin2A+2Cos2A=2Sin2A+Cos2A=2=Ruas Kanan 7.Cosec ACosec A-Sin A=Cot2A Ruas Kiri =1Sin A1Sin A-Sin A=1Sin2A-1=Cosec2A-1=Cot2A=Ruas Kanan
8.Sin4A-Cos4A=2Sin2A-1 Ruas Kiri =Sin4A-Cos4A=(Sin2A+Cos2A)Sin2A-Cos2A=(Sin2A+1-Sin2A)Sin2A1+Sin2A=2Sin2A-1=Ruas Kanan 9.Sec ASin A+Cosec ACos A=2Sin A.Cos A Ruas Kiri =Sec ASin A+Cosec ACos A=Sec A.Cos A+Cosec A.Sin ASin A.Cos A=2Sin A.Cos A=Ruas Kanan 10.Tan2A-Sin2A=Tan2A.Sin2A Ruas Kiri =Tan2A-Sin2A=Sin2ACos2A-Sin2A=Sin2A-Cos2A.Sin2ACos2A=Sin2A(1Cos2A)Cos2A=Tan2A.Sin2A=Ruas Kanan 11.Sec2A+Cot2A=Tan2A+Cosec2A Ruas Kiri =Sec2A+Cot2A=(1+Tan2A)+Cosec2A-1=Tan2A+Cosec2A=Ruas Kanan 12.Tan A+Cos A1+Sin A=Sec A Ruas Kiri =Tan A+Cos A1+Sin A=Sin ACos A+Cos A1+Sin A=1+Sin ASin A+Cos2ACos A(1+Sin A)=Sin A+Sin2A+Cos2ACos A(1+Sin A)=Sin A+1Cos A(1+Sin A)=1Cos A=Sec A=Ruas Kanan 13. 2Cosec A=Sin A1+Cos A+1+Cos ASin A Ruas Kanan=Sin A1+Cos A+1+Cos ASin A=Sin2ASin A(1+Cos A)+(1+Cos A)2Sin A(1+Cos A)=Sin2A+1+2Cos A+Cos2ASin A(1+Cos A)=2+2Cos ASin A(1+Cos A)=2(1+Cos A)Sin A(1+Cos A)=2Sin A=2Cosec A=Ruas Kiri 14.(Cos2A-1)Tan2A+1=1-Sec2A Ruas Kiri =(Cos2A-1)Tan2A+1=(1-Sin2A-1)Sec2A=-Sin2ACos2A=-Tan2A=-Sec2A1=1-Sec2A=Ruas Kanan 15.Sec ASin A+Cosec ACos A=2Sin A.Cos A Ruas Kiri =Sec ASin A+Cosec ACos A=Sec A.Cos A+Cosec A.Sin ASin A.Cos A=1+1Sin A.Cos A=2Sin A.Cos A=Ruas Kanan
1. Sin A1-Sin2A+Cos A1-Cos2A=Sin3A+Cos3ASin2A.Cos2A Ruas Kiri =Sin A1-Sin2A+Cos A1-Cos2A=Sin ACos2A+Cos ASin2A=Sin A.Sin2ACos2A.Sin2A+Cos A.Cos2ASin2A.Cos2A=Sin3A+Cos3ASin2A.Cos2A=Ruas Kanan 2. 1-2Cos2ASin A.Cos A=Tan A-Cot A Ruas Kiri =1-2Cos2ASin A.Cos A=Sin2A+Cos2A-2Cos2A)Sin A.Cos A=Sin2ACos2ASin A.Cos A=Sin ACos A-Cos ASin A=Tan A-Cot A=Ruas Kanan 3.1+Sec ATan A-Tan ASec A=1+Sec ASec A.Tan A Ruas Kiri = 1+Sec ATan A-Tan ASec A=Sec A+Sec2A-Tan2ATan A.Sec A=Sec A+1+Tan2A-Tan2ATan A.Sec A=1+Sec ASec A.Tan A=Ruas Kanan Tan A.Cot A+Tan A=1+Tan2A= Sec2A 4. Tan A-Sin ASin3A=Sec A1+Cos A Ruas Kiri =Tan A-Sin ASin3A=Tan A-Sin ASin A. Sin2A=Sin ACosA-Sin ASin A. Sin2A=1Cos A.Sin2A-1Sin2A=1-Cos ACos A.Sin2A=1-Cos ACos A.(1-Cos2A)=1Cos ACos A.(1-Cos A)(1+Cos A)=1Cos A.(1+Cos A)=1Cos A.1(1+Cos A)=Sec A1+Cos A=Ruas Kanan 5. Sin3A+Cos2ACosec A=Sin A Ruas Kiri = Sin3A+Cos2ACosec A=Sin3A+Sin A.Cos2A=Sin ASin2A+Cos2A=Sin A=Ruas Kanan 6.(Sin A-Cos A)2+(Sin A+Cos A)2=2 Ruas Kiri =(Sin A-Cos A)2+(Sin A+Cos A)2=Sin2A-2Sin A.Cos A+Cos2A+Sin2A+2Sin A.Cos A+Cos2A=2Sin2A+2Cos2A=2Sin2A+Cos2A=2=Ruas Kanan 7.Cosec ACosec A-Sin A=Cot2A Ruas Kiri =1Sin A1Sin A-Sin A=1Sin2A-1=Cosec2A-1=Cot2A=Ruas Kanan
8.Sin4A-Cos4A=2Sin2A-1 Ruas Kiri =Sin4A-Cos4A=(Sin2A+Cos2A)Sin2A-Cos2A=(Sin2A+1-Sin2A)Sin2A1+Sin2A=2Sin2A-1=Ruas Kanan 9.Sec ASin A+Cosec ACos A=2Sin A.Cos A Ruas Kiri =Sec ASin A+Cosec ACos A=Sec A.Cos A+Cosec A.Sin ASin A.Cos A=2Sin A.Cos A=Ruas Kanan 10.Tan2A-Sin2A=Tan2A.Sin2A Ruas Kiri =Tan2A-Sin2A=Sin2ACos2A-Sin2A=Sin2A-Cos2A.Sin2ACos2A=Sin2A(1Cos2A)Cos2A=Tan2A.Sin2A=Ruas Kanan 11.Sec2A+Cot2A=Tan2A+Cosec2A Ruas Kiri =Sec2A+Cot2A=(1+Tan2A)+Cosec2A-1=Tan2A+Cosec2A=Ruas Kanan 12.Tan A+Cos A1+Sin A=Sec A Ruas Kiri =Tan A+Cos A1+Sin A=Sin ACos A+Cos A1+Sin A=1+Sin ASin A+Cos2ACos A(1+Sin A)=Sin A+Sin2A+Cos2ACos A(1+Sin A)=Sin A+1Cos A(1+Sin A)=1Cos A=Sec A=Ruas Kanan 13. 2Cosec A=Sin A1+Cos A+1+Cos ASin A Ruas Kanan=Sin A1+Cos A+1+Cos ASin A=Sin2ASin A(1+Cos A)+(1+Cos A)2Sin A(1+Cos A)=Sin2A+1+2Cos A+Cos2ASin A(1+Cos A)=2+2Cos ASin A(1+Cos A)=2(1+Cos A)Sin A(1+Cos A)=2Sin A=2Cosec A=Ruas Kiri 14.(Cos2A-1)Tan2A+1=1-Sec2A Ruas Kiri =(Cos2A-1)Tan2A+1=(1-Sin2A-1)Sec2A=-Sin2ACos2A=-Tan2A=-Sec2A1=1-Sec2A=Ruas Kanan 15.Sec ASin A+Cosec ACos A=2Sin A.Cos A Ruas Kiri =Sec ASin A+Cosec ACos A=Sec A.Cos A+Cosec A.Sin ASin A.Cos A=1+1Sin A.Cos A=2Sin A.Cos A=Ruas Kanan
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