 New Zealand
Level 8 - NCEA Level 3

Solve equations using trigonometric identities

Lesson

Products of sine and cosines of two angles can be expressed as sums, and sums can be expressed as products.

The starting point is the angle sum formula,
$\sin(A+B)=\sin A\cos B+\sin B\cos A\ \ \ \ \ \ \ \ \ (1)$sin(A+B)=sinAcosB+sinBcosA         (1).

From this we deduce

 $\sin(A-B)$sin(A−B) $=$= $\sin A\cos(-B)+\sin(-B)\cos A$sinAcos(−B)+sin(−B)cosA $=$= $\sin A\cos B-\sin B\cos A\ \ \ \ \ \ \ \ (2)$sinAcosB−sinBcosA        (2)

For the cosines of a sum and difference, we have

 $\cos(A-B)$cos(A−B) $=$= $\sin\left(\frac{\pi}{2}-(A-B)\right)$sin(π2​−(A−B)) $=$= $\sin\left((\frac{\pi}{2}-A)+B\right)$sin((π2​−A)+B) $=$= $\sin(\frac{\pi}{2}-A)\cos B+\sin B\cos(\frac{\pi}{2}-A)$sin(π2​−A)cosB+sinBcos(π2​−A) $=$= $\cos A\cos B+\sin B\sin A\ \ \ \ \ (3)$cosAcosB+sinBsinA     (3)

And, from this we deduce

 $\cos(A+B)$cos(A+B) $=$= $\cos\left(A-(-B)\right)$cos(A−(−B)) $=$= $\cos A\cos(-B)+\sin(-B)\sin A$cosAcos(−B)+sin(−B)sinA $=$= $\cos A\cos B-\sin B\sin A\ \ \ \ \ \ \ \ (4)$cosAcosB−sinBsinA        (4)

Now, by adding equations $(1)$(1) and $(2)$(2), we get
$\sin(A+B)+\sin(A-B)=2\sin A\cos B$sin(A+B)+sin(AB)=2sinAcosB

That is, a product has been expressed as a sum:

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

Similarly, by adding equations $(3)$(3) and $(4)$(4) we get
$\cos(A-B)+\cos(A+B)=2\cos A\cos B$cos(AB)+cos(A+B)=2cosAcosB.

That is:

$\cos A\cos B=\frac{1}{2}\left(\cos(A-B)+\cos(A+B)\right)\ \ \ \ \ \ \ \ \ \ (6)$cosAcosB=12(cos(AB)+cos(A+B))          (6).

By subtracting equation $(4)$(4) from equation $(3)$(3), we have
$\cos(A-B)-\cos(A+B)=2\sin B\sin A$cos(AB)cos(A+B)=2sinBsinA.

That is:

$\sin A\sin B=\frac{1}{2}\left(\cos(A-B)-\cos(A+B)\right)\ \ \ \ \ \ \ \ \ \ (7)$sinAsinB=12(cos(AB)cos(A+B))          (7)

If these products can be expressed as sums, it must be possible to express sums as products. In each of the formulae $(5)$(5), $(6)$(6) and $(7)$(7), we can set $A+B=X$A+B=X and $A-B=Y$AB=Y. Then, we must have $A=\frac{X+Y}{2}$A=X+Y2 and $B=\frac{X-Y}{2}$B=XY2.

After substituting for $A$A and $B$B in the three equations, we obtain:

$\sin X+\sin Y=2\sin\frac{X+Y}{2}\cos\frac{X-Y}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)$sinX+sinY=2sinX+Y2cosXY2              (8)

$\cos X+\cos Y=2\cos\frac{X+Y}{2}\cos\frac{X-Y}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)$cosX+cosY=2cosX+Y2cosXY2              (9)

$\cos Y-\cos X=2\sin\frac{X+Y}{2}\sin\frac{X-Y}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)$cosYcosX=2sinX+Y2sinXY2              (10)

From $(8)$(8) we can argue that

$\sin X-\sin Y=\sin X+\sin(-Y)=2\sin\frac{X-Y}{2}\cos\frac{X+Y}{2}\ \ \ \ \ \ \ \ \ (11)$sinXsinY=sinX+sin(Y)=2sinXY2cosX+Y2         (11)

Example 1

Verify that $\frac{\sin(7\theta)+\sin(5\theta)}{\cos(7\theta)-\cos(5\theta)}=-\cot\theta$sin(7θ)+sin(5θ)cos(7θ)cos(5θ)=cotθ

Using formulae $(8)$(8) and $(10)$(10) we see that the left-hand side can be written

$\frac{2\sin\left(\frac{7\theta+5\theta}{2}\right)\cos\left(\frac{7\theta-5\theta}{2}\right)}{2\sin\left(\frac{7\theta+5\theta}{2}\right)\sin\left(\frac{5\theta-7\theta}{2}\right)}$2sin(7θ+5θ2)cos(7θ5θ2)2sin(7θ+5θ2)sin(5θ7θ2)

$=\frac{\cos\theta}{-\sin\theta}$=cosθsinθ

$=-\cot\theta$=cotθ

Example 2

Solve $\cos3x+\cos2x=0$cos3x+cos2x=0.

Formula $(9)$(9) is applicable. The equation can be re-written as $2\cos\frac{3x+2x}{2}\cos\frac{3x-2x}{2}=0$2cos3x+2x2cos3x2x2=0.

Thus, $2\cos\frac{5x}{2}\cos\frac{x}{2}=0$2cos5x2cosx2=0.

This has solutions where $\cos\frac{5x}{2}=0$cos5x2=0 and where $\cos\frac{x}{2}=0$cosx2=0.

So, $\frac{5x}{2}=\frac{\pi}{2}+2n\pi$5x2=π2+2nπ or $\frac{5x}{2}=\frac{3\pi}{2}+2n\pi$5x2=3π2+2nπ or $\frac{x}{2}=\frac{\pi}{2}+2n\pi$x2=π2+2nπ or $\frac{x}{2}=\frac{3\pi}{2}+2n\pi$x2=3π2+2nπ.

Therefore, the general solutions are
$x=\frac{4n+1}{5}\pi$x=4n+15π
$x=\frac{4n+3}{5}\pi$x=4n+35π
$x=(4n+1)\pi$x=(4n+1)π
$x=(4n+3)\pi$x=(4n+3)π

where $n$n is an integer.

In fact, we only need the first two solution expressions since the others are included in them. The graph of the function $y=\cos3x+\cos2x$y=cos3x+cos2x should help to visualise the locations of the zeros. Worked Examples

Question 1

Use the product-to-sum identities to express $2\cos\left(3x\right)\cos\left(9x\right)$2cos(3x)cos(9x) 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

Evaluate $\cos195^\circ-\cos75^\circ$cos195°cos75° by first expressing it as a product of two trigonometric functions.

Outcomes

M8-6

Manipulate trigonometric expressions

91575

Apply trigonometric methods in solving problems