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Published byMerilyn Hawkins Modified about 1 year ago

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COMPOUND ANGLE FORMULAE

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It can be shown that the compound angle formula for sin (A + B) is: Consider the expression: sin (A + B). Firstly note that sin (A + B) ≠ sin A + sin B ( This can easily be shown, e.g. let A = 30° and B = 60°). By letting B = – B in (1) sin (A – B) = sinA cos(–B) + cosA sin(–B)... (1) sin (A + B) = sinA cosB + cosA sinB... (2) sin (A – B) = sinA cosB – cosA sinB Since: sin(–B) = – sinB cos(–B) = cosB

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Let A = 90 – A: sin (90 – A + B) = sin(90 – A) cosB + cos(90 –A) sinB sin (90 – (A – B)) = sin(90 – A) cosB + cos(90 –A) sinB Let B = – B in (3): cos (A + B) = cosA cos(–B) + sinA sin(–B) Since:... (4) cos (A + B) = cosA cosB – sinA sinB... (3) cos (A – B) = cosA cosB + sinA sinBFrom (1): sin (A + B) = sinA cosB + cosA sinB sin(90 – A) = cosA cos(90 – A) = sinA

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sin (A + B) = sinA cosB + cosA sinB... (1) cos (A + B) = cosA cosB – sinA sinB... (4) Now let B = – B: tan (A + B) = sinA cosB + cosA sinB cosA cosB – sinA sinB Divide (1) by (4): Divide each term by cosA cosB tan (A + B) = sinA cosB cosA sinB cosA cosB sinA sinB cosA cosB – +... (5) tan (A + B) = tanA + tanB 1 – tanA tanB... (6) tan (A – B) = tanA – tanB 1 + tanA tanB

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Example 2: Find in surd form cos15°. Let A = 45° and B = 30° cos (45 – 30) = cos45 cos30 + sin45 sin30 1212 2 = 2 + 2 + 4 Example 1: Simplify sin(x + 90°). sin (x + 90) =sin x cos 90+ cos x sin 90 = sin x ( 0 )+ cos x ( 1 )= cos x 4 = + 4 = sin (A + B) = sinA cosB + cosA sinB Using:cos (A – B) = cosA cosB + sinA sinB Using:

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Example 3: Show that, in surd form tan75° = 2 + tan (45 + 30) = tan45 + tan30 1 – tan45 tan30 = 1 1 + 1 1 – 1 × = + 1 – 1 = 4 + 2 2 = 2 + tan (A + B) = tanA + tanB 1 – tanA tanB Using: = 3 + 2+ 1 3 – 1 Multiply each term by: × + 1 – 1 + 1 = (Rationalise here)

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Example 4: Solve the equation sin (x + 30°) = 2 cos (x + 60°) ; for 0 < x < 360°. sin (x + 30°) = 2 cos (x + 60°) = 2{ cos x cos 60 – sin x sin 60 } sin x 2 + cos x 1212 = 2 { 1212 sin x– 2 } 2 cos xsin xcos x + = sin x– 2 cos x 3 1 tan x = Now, TanA is positive in the 1 st and 3 rd quadrants: α = 10.89° α = tan -1 3 1 x = 10.9°, 190.9 ° Multiply by 2: sin x cos 30 + cos x sin 30 = 3sin x (1d.p.)

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Example 5: Prove the identity cot (A + B) = cot A + cot B cot A cot B – 1 cot (A + B) = LHS = tan (A + B) 1 tan A + tan B 1 – tan A tan B = tan (A + B) = tanA + tanB 1 – tanA tanB Divide each term by tan A tan B tan B 1 – 1 tan A 1 + tan A tan B 1 = cot B + cot A cot A cot B – 1 = = RHS

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Summary of key points: This PowerPoint produced by R.Collins ; Updated Mar. 2010 sin (A ± B) = sinA cosB ± cosA sinB cos (A ± B) = cosA cosB sinA sinB ± ± tan (A ± B) = tanA ± tanB 1 tanA tanB sin(90 – A) = cosA cos(90 – A) = sinA The compound angle formulae are: sin(–B) = – sinB cos(–B) = cosB The following results also need to be known:

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