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Trigonometry Examples
sin(2x)sin(2x)
Step 1
A good method to expand sin(2x)sin(2x) is by using De Moivre's theorem (r(cos(x)+i⋅sin(x))n=rn(cos(nx)+i⋅sin(nx)))(r(cos(x)+i⋅sin(x))n=rn(cos(nx)+i⋅sin(nx))). When r=1r=1, cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))ncos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n.
cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))ncos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n
Step 2
Expand the right hand side of cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))ncos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n using the binomial theorem.
Expand: (cos(x)+i⋅sin(x))2(cos(x)+i⋅sin(x))2
Step 3
Rewrite (cos(x)+isin(x))2(cos(x)+isin(x))2 as (cos(x)+isin(x))(cos(x)+isin(x))(cos(x)+isin(x))(cos(x)+isin(x)).
(cos(x)+isin(x))(cos(x)+isin(x))(cos(x)+isin(x))(cos(x)+isin(x))
Step 4
Step 4.1
Apply the distributive property.
cos(x)(cos(x)+isin(x))+isin(x)(cos(x)+isin(x))cos(x)(cos(x)+isin(x))+isin(x)(cos(x)+isin(x))
Step 4.2
Apply the distributive property.
cos(x)cos(x)+cos(x)(isin(x))+isin(x)(cos(x)+isin(x))cos(x)cos(x)+cos(x)(isin(x))+isin(x)(cos(x)+isin(x))
Step 4.3
Apply the distributive property.
cos(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
cos(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
Step 5
Step 5.1
Simplify each term.
Step 5.1.1
Multiply cos(x)cos(x)cos(x)cos(x).
Step 5.1.1.1
Raise cos(x)cos(x) to the power of 11.
cos1(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos1(x)cos(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
Step 5.1.1.2
Raise cos(x)cos(x) to the power of 11.
cos1(x)cos1(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos1(x)cos1(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
Step 5.1.1.3
Use the power rule aman=am+naman=am+n to combine exponents.
cos(x)1+1+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos(x)1+1+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
Step 5.1.1.4
Add 11 and 11.
cos2(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))cos2(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
cos2(x)+cos(x)(isin(x))+isin(x)cos(x)+isin(x)(isin(x))
Step 5.1.2
Rewrite using the commutative property of multiplication.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+isin(x)(isin(x))
Step 5.1.3
Multiply isin(x)(isin(x)).
Step 5.1.3.1
Raise i to the power of 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i1isin(x)sin(x)
Step 5.1.3.2
Raise i to the power of 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i1i1sin(x)sin(x)
Step 5.1.3.3
Use the power rule aman=am+n to combine exponents.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i1+1sin(x)sin(x)
Step 5.1.3.4
Add 1 and 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2sin(x)sin(x)
Step 5.1.3.5
Raise sin(x) to the power of 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2(sin1(x)sin(x))
Step 5.1.3.6
Raise sin(x) to the power of 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2(sin1(x)sin1(x))
Step 5.1.3.7
Use the power rule aman=am+n to combine exponents.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2sin(x)1+1
Step 5.1.3.8
Add 1 and 1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2sin2(x)
cos2(x)+icos(x)sin(x)+isin(x)cos(x)+i2sin2(x)
Step 5.1.4
Rewrite i2 as -1.
cos2(x)+icos(x)sin(x)+isin(x)cos(x)-1sin2(x)
Step 5.1.5
Rewrite -1sin2(x) as -sin2(x).
cos2(x)+icos(x)sin(x)+isin(x)cos(x)-sin2(x)
cos2(x)+icos(x)sin(x)+isin(x)cos(x)-sin2(x)
Step 5.2
Reorder the factors of isin(x)cos(x).
cos2(x)+icos(x)sin(x)+icos(x)sin(x)-sin2(x)
Step 5.3
Add icos(x)sin(x) and icos(x)sin(x).
cos2(x)+2icos(x)sin(x)-sin2(x)
cos2(x)+2icos(x)sin(x)-sin2(x)
Step 6
Move -sin2(x).
cos2(x)-sin2(x)+2icos(x)sin(x)
Step 7
Apply the cosine double-angle identity.
cos(2x)+2icos(x)sin(x)
Step 8
Step 8.1
Add parentheses.
cos(2x)+2i(cos(x)sin(x))
Step 8.2
Reorder 2i and cos(x)sin(x).
cos(2x)+cos(x)sin(x)(2i)
Step 8.3
Add parentheses.
cos(2x)+cos(x)(sin(x)⋅2)i
Step 8.4
Reorder cos(x) and sin(x)⋅2.
cos(2x)+sin(x)⋅2cos(x)i
Step 8.5
Reorder sin(x) and 2.
cos(2x)+2⋅sin(x)cos(x)i
Step 8.6
Apply the sine double-angle identity.
cos(2x)+sin(2x)i
cos(2x)+sin(2x)i
Step 9
Reorder factors in cos(2x)+sin(2x)i.
cos(2x)+isin(2x)
Step 10
Take out the expressions with the imaginary part, which are equal to sin(2x). Remove the imaginary number i.
sin(2x)=2sin(x)cos(x)