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Matematika - Identitas Trigonometri

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Matematika - Identitas Trigonometri

1. 1. y Consider the figure beside r y  sin   yO x x r y  r sin  1 x cos   r x  r cos  2 and : x 2  y 2  r 2 r cos  2  r sin  2  r 2 r 2 cos 2   r 2 sin   r 2 : r2 cos 2   sin 2   1
2. 2. cos 2   sin 2   1cos 2   1  sin 2  The formula sin 2   1  cos 2  ytan   x r sin tan   r cos  sin tan   cos 
3. 3. From the formula cos 2   sin 2   1 dividing by sin 2  cos 2  1 1  sin 2  sin 2  cot an2  1  cos ec 2 cos 2   sin 2   1 dividing by cos 2  sin 2  1 1  cos  cos 2  2 1  tan2   sec2 
4. 4. Examples: 1. Show that cos x  2 sin x   2 cos x  sin x   5 2 2 Solution: cos x  2 sin x2  2 cos x  sin x2  5cos 2 x  4 sin x cos x  4 sin 2 x  4 cos 2 x  4 sin x cos x  sin 2 x  5 5 cos 2 x  5 sin 2 x  5   5 cos 2 x  sin 2 x  5 55
5. 5. 2. Pr ove that : sin 4 x  cos 4 x  1  2 sin 2 x cos 2 xSolution: sin 4 x  cos 4 x  1  2 sin 2 x cos 2 x sin 2  2 x  cos x  2 sin 2 x cos 2 x  1  2 sin 2 x cos 2 x 2 1  2 sin 2 x cos 2 x  1  2 sin 2 x cos 2 x
6. 6. Activity classProve the following identities sec x. cos x 1. cot x  tan x 2. 3 cos x  4 sin x 2  4 cos x  3 sin x 2  25 3. a cos x  b sin x 2  b cos x  a sin x 2  a 2  b 2 4. tan x  cot anx  cos ecx. sec x cos x sin x 1 5.   cos x  sin x cos x  sin x 1  tan 2 x 6. cot B  tan B  cos ecB sec B