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 Skip Navigation LinksMath Help > Trigonometry > Trig Equivalences > Sin or Cos 3x, 4x, etc. > Sin or Cos 3x, 4x, etc.

Sine of multiples of x, in terms of cos x

As you know from Trig Equiv and Sin or Cos 3x,

sin 2x = 2 sin x cos x, 
sin 3x = -sin3x + 3cos2x sin x

If you keep going, you'll see the sines of all the multiples of x have only odd powers of sin x, so you can always use
sin2x=(1-cos2x) to express sin nx in terms sin x times an expression containing cos x:

sin 2x = (sin x) (2 cos x) 
sin 3x = (sin x) (-1 + 4cos2x)

As the multiples of x get higher, it gets harder and harder to multiply out all the factors of (1-cos2x), so I looked for a way to express cos nx in terms of cos x, cos (n-1)x, and cos (n-2)x.

Let y = (n-1)x.  Then nx is y+x and (n-2)x is y-x.

You know that

sin y+x = sin y cos x + cos y sin x, and
sin y-x = sin y cos x - cos y sin x

So the sum  of sin y+x and sin y-x is 2 sin y cos x.  So we get this useful trig equivalence:

sin y+x = 2 sin y cos x - sin y-x

Substituting (n-1)x in place of y, nx in place of y+x, and (n-2)x in place of y-x, we get this:

sin nx = 2 sin[(n-1)x] cos x - sin[(n-2)x]

Now that we know cos 2x and cos 3x, let's see how to use this fact to find cos 4x:

sin 2x = (sin x) (2 cos x) 
sin 3x = (sin x) (-1 + 4cos2x)
sin 4x = 2 sin 3x cos x - sin 2x
sin 4x = (sin x) ( 2 (-1 + 4cos2x)(cos x) - (2 cos x) )
sin 4x = (sin x) ( (-2 cos x + 8cos3x) - (2 cos x) )
sin 4x = (sin x) (-4 cos x + 8cos3x)

I'm far too lazy to keep doing these by hand, so I wrote a spreadsheet that calculates cos of all the multiples of x:

sin 1x = (sin) (1)
sin 2x = (sin) (2cos)
sin 3x = (sin) (-1+4cos2)
sin 4x = (sin) (-4cos+8cos3)
sin 5x = (sin) (1-12cos2+16cos4)
sin 6x = (sin) (6cos-32cos3+32cos5)
sin 7x = (sin) (-1+24cos2-80cos4+64cos6)
sin 8x = (sin) (-8cos+80cos3-192cos5+128cos7)
sin 9x = (sin) (1-40cos2+240cos4-448cos6+256cos8)
sin 10x = (sin) (10cos-160cos3+672cos5-1024cos7+512cos9)
sin 11x = (sin) (-1+60cos2-560cos4+1792cos6-2304cos8+1024cos10)
sin 12x = (sin) (-12cos+280cos3-1792cos5+4608cos7-5120cos9+2048cos11)
sin 13x = (sin) (1-84cos2+1120cos4-5376cos6+11520cos8-11264cos10+4096cos12)
sin 14x = (sin) (14cos-448cos3+4032cos5-15360cos7+28160cos9-24576cos11+8192cos13)
sin 15x = (sin) (-1+112cos2-2016cos4+13440cos6-42240cos8+67584cos10-53248cos12+16384cos14)
sin 16x = (sin) (-16cos+672cos3-8064cos5+42240cos7-112640cos9+159744cos11-114688cos13+32768cos15)
sin 17x = (sin) (1-144cos2+3360cos4-29568cos6+126720cos8-292864cos10+372736cos12-245760cos14+65536cos16)
sin 18x = (sin) (18cos-960cos3+14784cos5-101376cos7+366080cos9-745472cos11+860160cos13-524288cos15+131072cos17)

How was this table created?

A good question.  I'm glad you asked!  The coefficients of these formulas form a kind of "Pascal's Triangle", except we use a slightly different calculation from that of Pascal's Triangle, which gives the binomial coefficients.

Each coefficient is the twice the one above and to the left of it minus the one two rows above it.  That formula comes directly from this equivalence, which we developed, above:

sin nx = 2 sin[(n-1)x] cos x - sin[(n-2)x]

I'll give you the first few rows of the triangle:

sin nx

1      
      2      
-1      4      
     -4       8      
1      -12        16      
      6      -32         32      
-1      24       -80          64      
      -8      80       -192          128      
1      -40       240         -448          256      
      10     -160      672         -1024          512      
-1      60       -560         1792        -2304         1024      
     -12     280       -1792       4608         -5120         2048      
1        -84      1120       -5376        11520       -11264         4096      
      14     -448       4032       -15360       28160        -24576         8192      
-1      112       -2016      13440       -42240      67584        -53248        16384      
     -16      672       -8064       42240       -112640      159744      -114688       32768      
1        -144      3360      -29568      126720      -292864      372736      -245760        65536      
      18      -960      14784      -101376      366080      -745472      860160      -524288      131072      

Related Pages in this website

Special Angles

Trig Equivalences

Sin or Cos 3x, 4x, etc. -- trig functions of any multiple of an angle.

Cos of multiples of x, in terms of cos x

d/dx (sin x) = cos x, in the calculus section of this website

Hyperbolic Functions -- sinh(x) and cosh(x), which, together with exp(x) and the circular functions sin(x) and cos(x) form a family of functions.

 

 


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