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18.03.2013

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R O. AOK - OA (|OA| = R) .


1 (1 ) - , , 1/360 . ∠ AOK = α, ∠ AOB = 90, ∠ AOC = 180, ∠ AOD = 270, ∠ AOA = 360. 360, 60 (60'), 60 (60").

. Ox ( ∠ AOM = -β - ).

: 1 - , , . 2πR, 360 = 2π .

1 = 360/ = 5717'44",

1 = /360 = π/180 .


R M(x; y), ∠ AOM = α, |OM| = R (. ). α - (sin α), (cos α), (tg α) (ctg α):

sin α = y/R, cos α = x/R, tg α = y/x, ctg α = x/y.

( M I, II, III IV).


:

sin α = a/c, cos α = b/c, tg α = a/b, ctg α = b/a,

a - , α, b - , α, c - .

:


:

1. sin2 α + cos2 α = 1;

2. tg α ctg α = 1;

3. 1 + tg2 α = 1/cos2 α; 1 + ctg2 α = 1/sin2 α;

4. sin (π/2 - α) = cos α; cos (π/2 - α) = sin α


.

1. cos (α + β) = cos α cos β - sin α sin β;

2. cos (α - β) = cos α cos β + sin α sin β;

3. sin (α + β) = sin α cos β + cos α sin β;

4. sin (α - β) = sin α cos β - cos α sin β;

5. tg (α + β) = (tg α + tg β)/(1 - tg α tg β);

6. tg (α - β) = (tg α - tg β)/(1 + tg α tg β).


, .

1. sin 2α = 2 sin α cos α = 2tg α/(1 + tg2 α);

2. cos 2α = cos2 α - sin2 α = 2cos2 α - 1 = (1 - tg2 α)/(1 + tg2 α);

3. tg 2α = 2tg α/(1 - tg2 α);

4. sin 3α = sin α (3 - 4sin2 α);

5. cos 3α = cos α (4 cos2 α - 3);

6. tg 3α = (3tg α - tg3 α)/(1 - 3tg2 α).


.

1. cos α + cos β = 2cos(α + β)/2 cos(α - β)/2;

2. cos α - cos β = -2sin(α + β)/2 sin(α - β)/2;

3. sin α + sin β = 2sin(α + β)/2 cos(α - β)/2;

4. sin α - sin β = 2cos(α + β)/2 sin(α - β)/2;

5. tg α + tg β = sin(α + β)/cos α cos β;

6. tg α - tg β = sin(α - β)/cos α cos β.


.

1. cos α cos β = ½ [cos(α - β) + cos(α + β)];

2. sin α sin β = ½ [cos(α - β) - cos(α + β)];

3. sin α cos β = ½ [sin(α + β) + sin(α - β)].


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