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Sum and Difference Formulas Double-Angle, and the Half-Angle Formulas,[object Object]
Difference-Sum of Angles Formulas,[object Object],–,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],S(A±B) = S(A)C(B) ± S(B)C(A),[object Object],Double-Angle Formulas,[object Object],Half-Angle Formulas,[object Object],S(2A) = 2S(A)C(A),[object Object],,[object Object],1 + C(B),[object Object],B,[object Object],±,[object Object],C(   ) =,[object Object],2,[object Object],2,[object Object],C(2A) = C2(A) – S2(A),[object Object],              = 2C2(A) – 1,[object Object],              = 1 – 2S2(A) ,[object Object], Frank Ma,[object Object],2006,[object Object],,[object Object],1 – C(B),[object Object],B,[object Object],±,[object Object],S(   ) =,[object Object],2,[object Object],2,[object Object]
Difference-Sum of Angles Formulas,[object Object],–,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],S(A±B) = S(A)C(B) ± S(B)C(A),[object Object],Double-Angle Formulas,[object Object],Half-Angle Formulas,[object Object],S(2A) = 2S(A)C(A),[object Object],,[object Object],1 + C(B),[object Object],B,[object Object],±,[object Object],C(   ) =,[object Object],2,[object Object],2,[object Object],C(2A) = C2(A) – S2(A),[object Object],              = 2C2(A) – 1,[object Object],              = 1 – 2S2(A) ,[object Object], Frank Ma,[object Object],2006,[object Object],,[object Object],1 – C(B),[object Object],B,[object Object],±,[object Object],S(   ) =,[object Object],2,[object Object],2,[object Object],The cosine-difference formula is the basis for all the other formulas listed above.,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],3π,[object Object],8π,[object Object],11π,[object Object],=,[object Object],+,[object Object],12,[object Object],12,[object Object],12,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Example A: Find cos(11π/12) without a calculator.,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Example A: Find cos(11π/12) without a calculator.,[object Object],11π,[object Object],π,[object Object],2π,[object Object],cos(      ) ,[object Object],=,[object Object],cos(              )   ,[object Object],+,[object Object],12,[object Object],4,[object Object],3,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Example A: Find cos(11π/12) without a calculator.,[object Object],11π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],cos(      ) ,[object Object],=,[object Object],cos(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],c(    ) ,[object Object],+,[object Object],s(    ) ,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Cosine-Sum Formulas,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Example A: Find cos(11π/12) without a calculator.,[object Object],11π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],cos(      ) ,[object Object],=,[object Object],cos(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],c(    ) ,[object Object],+,[object Object],s(    ) ,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],2,[object Object],2,[object Object],3,[object Object],(-1),[object Object],=,[object Object],– ,[object Object],Cosine-Sum Formulas,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object]
Cosine-Sum-Difference Formulas,[object Object],cos(A + B) = cos(A)cos(B) – sin(A)sin(B),[object Object],cos(A – B) = cos(A)cos(B) + sin(A)sin(B),[object Object],–,[object Object],Short version:,[object Object],C(A±B) = C(A)C(B)    S(A)S(B),[object Object],+,[object Object],All fractions with denominator 12 may be written as sum or difference of fractions with denominators ,[object Object],3, 6 and 4.,[object Object],π,[object Object],π,[object Object],π,[object Object],3π,[object Object],8π,[object Object],π,[object Object],2π,[object Object],11π,[object Object],=,[object Object],=,[object Object],– ,[object Object],+,[object Object],=,[object Object],+,[object Object],;,[object Object],12,[object Object],12,[object Object],12,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Example A: Find cos(11π/12) without a calculator.,[object Object],11π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],π,[object Object],2π,[object Object],cos(      ) ,[object Object],=,[object Object],cos(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],c(    ) ,[object Object],+,[object Object],s(    ) ,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],2,[object Object],2,[object Object],3,[object Object],-2 – 6,[object Object],(-1),[object Object], -0.966,[object Object],=,[object Object],– ,[object Object],=,[object Object],Cosine-Sum Formulas,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],4,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)),[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, ,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object],Example B: Find sin(– π/12) without a calculator.,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object],Example B: Find sin(– π/12) without a calculator.,[object Object],–π,[object Object],π,[object Object],π,[object Object],sin(      ) ,[object Object],=,[object Object],sin(              )   ,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object],Example B: Find sin(– π/12) without a calculator.,[object Object],–π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],sin(      ) ,[object Object],=,[object Object],sin(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],s(    ) ,[object Object],c(    ) ,[object Object],–,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],Sum Formulas,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object],Example B: Find sin(– π/12) without a calculator.,[object Object],–π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],sin(      ) ,[object Object],=,[object Object],sin(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],s(    ) ,[object Object],c(    ) ,[object Object],–,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],2,[object Object],2,[object Object],3,[object Object],1,[object Object],=,[object Object],– ,[object Object],=,[object Object],Sum Formulas,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object]
Sine-Sum-Difference Formulas,[object Object],From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: ,[object Object],sin(A + B) = sin(A)cos(B) + cos(A)sin(B),[object Object],Write sin(A – B) = sin(A + (-B)), expand we get:  ,[object Object],sin(A – B) = sin(A)cos(B) – cos(A)sin(B),[object Object],Short version:,[object Object],S(A±B) = S(A)C(B) ± C(A)S(B),[object Object],Example B: Find sin(– π/12) without a calculator.,[object Object],–π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],π,[object Object],sin(      ) ,[object Object],=,[object Object],sin(              )   ,[object Object],c(     ) ,[object Object],s(     ) ,[object Object],=,[object Object],s(    ) ,[object Object],c(    ) ,[object Object],–,[object Object],–,[object Object],12,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],4,[object Object],3,[object Object],2,[object Object],2,[object Object],3,[object Object],2 – 6,[object Object],1,[object Object], -0.259,[object Object],=,[object Object],– ,[object Object],=,[object Object],Sum Formulas,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],4,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) ,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)),[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)) ,[object Object],= 2cos2(A) – 1,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)) ,[object Object],= 2cos2(A) – 1,[object Object],sin(2A) = sin(A + A) ,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)) ,[object Object],= 2cos2(A) – 1,[object Object],sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A),[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)) ,[object Object],= 2cos2(A) – 1,[object Object],sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A),[object Object],sin(2A) = 2sin(A)cos(A) ,[object Object]
Double Angle Formulas,[object Object],From the sum-of-angle formulas, we obtain the ,[object Object],double-angle formulas by setting A = B shown here,,[object Object],cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A),[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],(1 – sin2(A)) – sin2(A),[object Object],= 1 – 2sin2(A),[object Object],cos2(A) –(1 – cos2(A)) ,[object Object],= 2cos2(A) – 1,[object Object],sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A),[object Object],sin(2A) = 2sin(A)cos(A) ,[object Object],Cosine Double Angle Formulas:,[object Object],Sine Double Angle Formulas:,[object Object],cos(2A) = cos2(A) – sin2(A) ,[object Object],             = 1 – 2sin2(A)         ,[object Object],             = 2cos2(A) – 1 ,[object Object],sin(2A) = 2sin(A)cos(A) ,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, ,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object],y,[object Object],A,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object],y2 + (-5)2 = (7)2,[object Object],y,[object Object],A,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object],y2 + (-5)2 = (7)2,[object Object],y2 = 2,[object Object],y,[object Object],A,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object],y2 + (-5)2 = (7)2,[object Object],y2 = 2,[object Object],y,[object Object],A,[object Object],y = ±2 ,[object Object],y = 2 ,[object Object]
Double Angle Formulas,[object Object],Example C:  Given angle A in the 2nd quad. and ,[object Object],cos(2A)= 3/7, find tan(A). ,[object Object],Use the formula cos(2A) = 2cos2(A) – 1, we get,[object Object],                             3/7 = 2cos2(A) – 1 ,[object Object],                            10/7 = 2cos2(A),[object Object],                              5/7 = cos2(A),[object Object],±5/7 = cos(A),[object Object],Since A is in 2nd quad.=> cos(A) = - 5/7 ,[object Object], Frank Ma,[object Object],2006,[object Object],y2 + (-5)2 = (7)2,[object Object],y2 = 2,[object Object],y,[object Object],A,[object Object],y = ±2 ,[object Object],y = 2 ,[object Object],2,[object Object],,[object Object],Therefore tan(A) = ,[object Object],–,[object Object], -0.632,[object Object],5,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object],1+cos(2A),[object Object],cos2(A) =,[object Object],2,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object],1+cos(2A),[object Object],cos2(A) =,[object Object],2,[object Object],In the square root form, we get,[object Object],,[object Object],1+cos(2A),[object Object],±,[object Object],cos(A) =,[object Object],2,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object],1+cos(2A),[object Object],cos2(A) =,[object Object],2,[object Object],In the square root form, we get,[object Object],,[object Object],1+cos(2A),[object Object],±,[object Object],cos(A) =,[object Object],2,[object Object],if we replace A by B/2 so that 2A = B, ,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object],1+cos(2A),[object Object],cos2(A) =,[object Object],2,[object Object],In the square root form, we get,[object Object],,[object Object],1+cos(2A),[object Object],±,[object Object],cos(A) =,[object Object],2,[object Object],if we replace A by B/2 so that 2A = B, we get the ,[object Object],half-angle formula of cosine:,[object Object],B,[object Object],,[object Object],1+cos(B),[object Object],±,[object Object],cos(   ) =,[object Object],2,[object Object],2,[object Object]
Half-angle Formulas,[object Object],From cos(2A) = 2cos2(A) – 1, we get ,[object Object],1+cos(2A),[object Object],cos2(A) =,[object Object],2,[object Object],In the square root form, we get,[object Object],,[object Object],1+cos(2A),[object Object],±,[object Object],cos(A) =,[object Object],2,[object Object],if we replace A by B/2 so that 2A = B, we get the ,[object Object],half-angle formula of cosine:,[object Object],B,[object Object],,[object Object],1+cos(B),[object Object],±,[object Object],cos(   ) =,[object Object],2,[object Object],2,[object Object],Similarly, we get the half-angle formula of sine: ,[object Object],B,[object Object],,[object Object],1 – cos(B),[object Object],±,[object Object],sin(   ) =,[object Object],2,[object Object],2,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],-7,[object Object],-3,[object Object],A,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],-7,[object Object],-3,[object Object],A,[object Object],58,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],Since –π < A < –π /2, so ,[object Object],–π/2 < A/2 < –π/4, ,[object Object],-7,[object Object],-3,[object Object],A,[object Object],58,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],Since –π < A < –π /2, so ,[object Object],–π/2 < A/2 < –π/4, we have that,[object Object],A/2 is in the 4th quadrant. ,[object Object],-7,[object Object],-3,[object Object],A,[object Object],58,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],Since –π < A < –π /2, so ,[object Object],–π/2 < A/2 < –π/4, we have that,[object Object],A/2 is in the 4th quadrant. ,[object Object],-7,[object Object],-3,[object Object],1 + cos(A),[object Object],,[object Object],A,[object Object],A,[object Object],cos(   ) =,[object Object],Hence,,[object Object],58,[object Object],2,[object Object],2,[object Object]
Half-angle Formulas,[object Object],,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],B,[object Object],±,[object Object],±,[object Object],1+cos(B),[object Object],cos(   ) =,[object Object],sin(   ) =,[object Object],and,[object Object],2,[object Object],2,[object Object],2,[object Object],2,[object Object],The ± are to be determined by the position of the ,[object Object],angle B/2. ,[object Object],Example D:  Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).,[object Object],Since –π < A < –π /2, so ,[object Object],–π/2 < A/2 < –π/4, we have that,[object Object],A/2 is in the 4th quadrant. ,[object Object],-7,[object Object],-3,[object Object],1 + cos(A),[object Object],,[object Object],A,[object Object],A,[object Object],cos(   ) =,[object Object],Hence,,[object Object],58,[object Object],2,[object Object],2,[object Object],,[object Object],1 – 7 /58,[object Object], 0.201,[object Object],=,[object Object],2,[object Object]
Sum of Angles Formulas,[object Object],±,[object Object],Double Angle Formulas,[object Object],Half Angle Formulas,[object Object],sin(2A) = 2sin(A)cos(A),[object Object],,[object Object],1+cos(B),[object Object],B,[object Object],±,[object Object],cos(   ) =,[object Object],2,[object Object],2,[object Object],cos(2A) = cos2(A) – sin2(A),[object Object],              = 2cos2(A) – 1,[object Object],              = 1 – 2sin2(A) ,[object Object], Frank Ma,[object Object],2006,[object Object],,[object Object],1 – cos(B),[object Object],B,[object Object],±,[object Object],sin(   ) =,[object Object],2,[object Object],2,[object Object]

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