Multiples And Sub-Multiples
Transformation :
Theorem : for all C,D\(
\in
\) R
1) sinC + sinD =\(
2sin\frac{{C + D}}
{2}.\cos \frac{{C - D}}
{2}
\)
2) sinC - sinD = \(
2sin\frac{{C - D}}
{2}.\cos \frac{{C + D}}
{2}
\)
3) cosC + cosD =\(
2\cos \frac{{C + D}}
{2}.\cos \frac{{C - D}}
{2}
\)
4) cosC - cosD =\(
- 2\sin \frac{{C + D}}
{2}.sin\frac{{C - D}}
{2}
\)
Proof : For all A,B \(
\in
\)|R, we have
sin(A+B) = sinA cosB + cosA . sinB
and sin (A - B) = sinA cosB - cosA . sinB
by adding sin(A+B) +sin(A-B) = 2sinA cosB......(1)
by subtracting , sin(A-B) - sin(A-B) =2cosA sinB ....(2)
for all A, B \(
\in
\) R, we have
cos(A+B) = cosA cosB - sinA sinB
cos(A-B) = cosA cosB +sinA sinB
by adding cos(A+B) +cos(A-B) = 2cosA cosB ......(3)
by subtracting , cos(A+B) - cos(A-B) =2sinA sinB ....(4)
put A+B =C, A-B=D, in the above 4 formule
\(
\Rightarrow A = \frac{{C + D}}
{2},\,\,\,\,\,\,B = \frac{{C - D}}
{2}
\) …...(5)
\(
\Rightarrow \sin C - \sin D = 2\cos \frac{{C + D}}
{2},\sin \frac{{C - D}}
{2}
\) …...(6)
\(
\Rightarrow \cos C + \cos D = 2\cos \frac{{C + D}}
{2}.\cos \frac{{C - D}}
{2}
\)…….(7)
\(
\Rightarrow \operatorname{cosC} - cosD = - 2sin\frac{{C + D}}
{2}.\sin \frac{{C - D}}
{2}
\)…….(8)
Condtional Identiites :
Where the angles A, B and C satisfy a given relation, many intersting identities can be established connecting the trigonometric functions of these the properties of complementary and supplementary angles for examples if A +B+C =\(
\pi
\)
1) sin(B+C) = sinA, cosB = - cos(C+A)
2) cos(A+B) = -cosC, sinC = sin(A+B)
3) tan(C+A) = - tanB, cotA = -cot(B+C)
4)\(
\cos \left( {\frac{{A + B}}
{2}} \right) = \sin \frac{C}
{2},\,\,\cos \frac{c}
{2} = \sin \left( {\frac{{A + B}}
{2}} \right)
\)
5)\(
\sin \left( {\frac{{C + A}}
{2}} \right) = \cos \frac{B}
{2},\,\,\sin \frac{A}
{2} = \cos \left( {\frac{{B + C}}
{2}} \right)
\)
6)\(
\tan \left( {\frac{{B + C}}
{2}} \right) = \cot \frac{A}
{2},\,\,\tan \frac{B}
{2} = \cot \left( {\frac{{C + A}}
{2}} \right)
\)