11 X1 T04 14 Products To Sums Etc
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11 X1 T04 14 Products To Sums Etc

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    11 X1 T04 14 Products To Sums Etc 11 X1 T04 14 Products To Sums Etc Presentation Transcript

    • Products to Sums
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B )
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B )
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B )
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x )
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x )
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x b) cos3θ cos5θ
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x 1 = ( 2cos3θ cos5θ ) b) cos3θ cos5θ 2
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x 1 = ( 2cos3θ cos5θ ) b) cos3θ cos5θ 2 1 = ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) ) 2
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x 1 = ( 2cos3θ cos5θ ) b) cos3θ cos5θ 2 1 = ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) ) 2 1 = ( cos8θ + cos ( −2θ ) ) 2
    • Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B ) 2cos A cos B = cos ( A + B ) + cos ( A − B ) 2sin A sin B = cos ( A − B ) − cos ( A + B ) eg (i) Express as a sum or difference of trig functions a ) 2cos5 x sin x = 2sin x cos5 x = sin ( x + 5 x ) + sin ( x − 5 x ) = sin 6 x + sin ( −4 x ) = sin 6 x − sin 4 x 1 = ( 2cos3θ cos5θ ) b) cos3θ cos5θ 2 1 = ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) ) 2 1 1 = ( cos8θ + cos 2θ ) = ( cos8θ + cos ( −2θ ) ) 2 2
    • (ii) Evaluate 2sin 45o cos15o
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) 2 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) (2 sine half sum cos half diff) 2 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) (2 sine half sum cos half diff) 2 2 1 1 cos A + cos B = 2cos ( A + B ) cos ( A − B ) 2 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) (2 sine half sum cos half diff) 2 2 1 1 cos A + cos B = 2cos ( A + B ) cos ( A − B ) (2 cos half sum cos half diff) 2 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) (2 sine half sum cos half diff) 2 2 1 1 cos A + cos B = 2cos ( A + B ) cos ( A − B ) (2 cos half sum cos half diff) 2 2 1 1 cos A − cos B = −2sin ( A + B ) sin ( A − B ) 2 2
    • (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2 Sums to Products 1 1 sin A + sin B = 2sin ( A + B ) cos ( A − B ) (2 sine half sum cos half diff) 2 2 1 1 cos A + cos B = 2cos ( A + B ) cos ( A − B ) (2 cos half sum cos half diff) 2 2 1 1 cos A − cos B = −2sin ( A + B ) sin ( A − B ) (minus 2 sine half sum 2 2 sine half diff)
    • eg (i) Convert into products of trig functions
    • eg (i) Convert into products of trig functions a ) cos3 A − cos5 A
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A )
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x )
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0 sin 2 x = 0 cos x = 0 or
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0 sin 2 x = 0 cos x = 0 or 2 x = 0o ,180o ,360o ,540o ,720o
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0 sin 2 x = 0 cos x = 0 or 2 x = 0o ,180o ,360o ,540o ,720o x = 0o ,90o ,180o , 270o ,360o
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0 sin 2 x = 0 cos x = 0 or x = 90o , 270o 2 x = 0o ,180o ,360o ,540o ,720o x = 0o ,90o ,180o , 270o ,360o
    • eg (i) Convert into products of trig functions 1 1 a ) cos3 A − cos5 A = −2sin ( 8 A ) sin ( −2 A ) 2 2 = −2sin 4 A sin ( − A ) = 2sin 4 A sin A b) sin 6 x − sin 4 x = sin 6 x + sin ( −4 x ) 1 1 = 2sin ( 2 x ) cos ( 10 x ) 2 2 = 2sin x cos5 x (ii) Solve sin x + sin 3 x = 0 0o ≤ x ≤ 360o 2sin 2 x cos ( − x ) = 0 2sin 2 x cos x = 0 sin 2 x = 0 cos x = 0 or x = 90o , 270o 2 x = 0o ,180o ,360o ,540o ,720o x = 0o ,90o ,180o , 270o ,360o ∴ x = 0o ,90o ,180o , 270o ,360o
    • Exercise 2F; 1b, 2b, 3a, 9, 10ace