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India
Class XI

Sums and differences as products (deg)

Lesson

Products

If the sine and cosine sum and difference formulas are written down side-by-side it becomes apparent that useful results can be obtained by adding some of them them in pairs.

$\sin\left(A+B\right)\equiv\sin A\cos B+\sin B\cos A$sin(A+B)sinAcosB+sinBcosA $\left(1\right)$(1)
$\sin\left(A-B\right)\equiv\sin A\cos B-\sin B\cos A$sin(AB)sinAcosBsinBcosA $\left(2\right)$(2)
$\cos\left(A+B\right)\equiv\cos A\cos B-\sin A\sin B$cos(A+B)cosAcosBsinAsinB $\left(3\right)$(3)
$\cos\left(A-B\right)\equiv\cos A\cos B+\sin A\sin B$cos(AB)cosAcosB+sinAsinB $\left(4\right)$(4)

 

 

 

 

If we add (1) and (2), we have 

$\sin\left(A+B\right)+\sin\left(A-B\right)=2\sin A\cos B$sin(A+B)+sin(AB)=2sinAcosB $\left(5\right)$(5)

Similarly, from (3) and (4) we obtain, by addition,

$\cos\left(A+B\right)+\cos\left(A-B\right)=2\cos A\cos B$cos(A+B)+cos(AB)=2cosAcosB $\left(6\right)$(6)

 

 

and by subtraction,

$\cos\left(A-B\right)-\cos\left(A+B\right)=2\sin A\sin B$cos(AB)cos(A+B)=2sinAsinB $\left(7\right)$(7)

Equations (5), (6) and (7) give the following three product formulas:

$\sin A\cos B=\frac{1}{2}\left(\sin\left(A+B\right)+\sin\left(A-B\right)\right)$sinAcosB=12(sin(A+B)+sin(AB)) $\left(5a\right)$(5a)
$\cos A\cos B=\frac{1}{2}\left(\cos\left(A+B\right)+\cos\left(A-B\right)\right)$cosAcosB=12(cos(A+B)+cos(AB)) $\left(6a\right)$(6a)
$$ $\left(7a\right)$(7a)

Sums

By re-writing  (5a), (6a) and (7a) we can obtain formulas for the sums and differences of sines and cosines. To do this, we let $U=A+B$U=A+B and $V=A-B$V=AB. Then, by solving these equations for $A$A and $B$B we get $A=\frac{U+V}{2}$A=U+V2 and $B=\frac{U-V}{2}$B=UV2

Thus, by substituting for $A$A and $B$B in the product formulas and rearranging slightly, we obtain:

$\sin U+\sin V=2\sin\frac{U+V}{2}\cos\frac{U-V}{2}$sinU+sinV=2sinU+V2cosUV2 $\left(8\right)$(8)
$\cos U+\cos V=2\cos\frac{U+V}{2}\cos\frac{U-V}{2}$cosU+cosV=2cosU+V2cosUV2 $\left(9\right)$(9)
$\cos V-\cos U=2\sin\frac{U+V}{2}\sin\frac{U-V}{2}$cosVcosU=2sinU+V2sinUV2 $\left(10\right)$(10)

 

 

 

 

and from (8), using the fact that $-\sin V=\sin\left(-V\right),$sinV=sin(V),we can write

$\sin U-\sin V=2\sin\frac{U-V}{2}\cos\frac{U+V}{2}$sinUsinV=2sinUV2cosU+V2 $\left(11\right)$(11)

 

 

Another type of sum, with a very useful simplification, occurs between different multiples of the sine and cosine of identical angles.

The expression $a\sin\theta+b\cos\theta$asinθ+bcosθ  can be written in the form $r\sin\left(\theta+\alpha\right)$rsin(θ+α). The latter expands to $r\left(\sin\theta\cos\alpha+\cos\theta\sin\alpha\right)$r(sinθcosα+cosθsinα).

On comparing this with the original expression, we see that $a=r\cos\alpha$a=rcosα and $b=r\sin\alpha$b=rsinα.

Hence, $r=\sqrt{a^2+b^2}$r=a2+b2 and $\tan\alpha=\frac{b}{a}$tanα=ba. Then, using the notation $\tan^{-1}$tan1 for the inverse tangent function, we can write

$$ $\left(12\right)$(12)

 

 

Example

Express $\cos255^\circ-\cos45^\circ$cos255°cos45° more simply.

Using (10), $\cos V-\cos U=2\sin\frac{U+V}{2}\sin\frac{U-V}{2}$cosVcosU=2sinU+V2sinUV2, we have 

$\cos255^\circ-\cos45^\circ=2\sin\frac{255^\circ+45^\circ}{2}\sin\frac{45^\circ-255^\circ}{2}$cos255°cos45°=2sin255°+45°2sin45°255°2

That is,

$\cos255^\circ-\cos45^\circ=$cos255°cos45°= $2\sin150^\circ\sin\left(-105\right)^\circ$2sin150°sin(105)°
$=$= $-2\sin30^\circ\sin75^\circ$2sin30°sin75°
$=$= $-\sin75^\circ$sin75°

Using a half-angle formula, $$ we can further simplify this to the exact value $-\frac{1}{2}\sqrt{2+\sqrt{3}}$122+3.

Worked Examples

QUESTION 1

Express $\cos\left(3x+2y\right)\cos\left(x-y\right)$cos(3x+2y)cos(xy) as a sum or difference of two trigonometric functions.

QUESTION 2

Express $\sin\left(6x\right)+\sin\left(4x\right)$sin(6x)+sin(4x) as a product of two trigonometric functions.

QUESTION 3

By using the product-to-sum identities, rewrite $2\sin53^\circ\cos116^\circ$2sin53°cos116° as a sum or difference of two trigonometric values.

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

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