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Popular Trigonometry >

(cos(pi/3)+isin(pi/3))^2

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Solution

(cos(3π​)+isin(3π​))2

Solution

−21​+i23​​
Solution steps
(cos(3π​)+isin(3π​))2
Use the following trivial identity:cos(3π​)=21​
cos(3π​)
cos(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​cos(x)123​​22​​21​0−21​−22​​−23​​​xπ67π​45π​34π​23π​35π​47π​611π​​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
=21​
Use the following trivial identity:sin(3π​)=23​​
sin(3π​)
sin(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​sin(x)021​22​​23​​123​​22​​21​​xπ67π​45π​34π​23π​35π​47π​611π​​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
=23​​
=(21​+i23​​)2
Simplify (21​+i23​​)2:−21​+i23​​
(21​+i23​​)2
Multiply i23​​:23​i​
i23​​
Multiply fractions: a⋅cb​=ca⋅b​=23​i​
=(21​+23​i​)2
Combine the fractions 21​+23​i​:21+3​i​
Apply rule ca​±cb​=ca±b​=21+3​i​
=(21+3​i​)2
Apply exponent rule: (ba​)c=bcac​=22(1+3​i)2​
(1+3​i)2=−2+23​i
(1+3​i)2
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2a=1,b=3​i
=12+2⋅1⋅3​i+(3​i)2
Simplify 12+2⋅1⋅3​i+(3​i)2:−2+23​i
12+2⋅1⋅3​i+(3​i)2
Apply rule 1a=112=1=1+2⋅1⋅3​i+(3​i)2
2⋅1⋅3​i=23​i
2⋅1⋅3​i
Multiply the numbers: 2⋅1=2=23​i
(3​i)2=−3
(3​i)2
Apply exponent rule: (a⋅b)n=anbn=i2(3​)2
(3​)2:3
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
=3i2
i2=−1
i2
Apply imaginary number rule: i2=−1=−1
=3(−1)
Remove parentheses: (−a)=−a=−3⋅1
Multiply the numbers: 3⋅1=3=−3
=1+23​i−3
Subtract the numbers: 1−3=−2=−2+23​i
=−2+23​i
=22(−2+23​i)​
Remove parentheses: (−a)=−a=22−2+23​i​
Factor −2+23​i:2(−1+3​i)
−2+23​i
Rewrite as=−2⋅1+23​i
Factor out common term 2=2(−1+3​i)
=222(−1+3​i)​
Cancel the common factor: 2=2−1+3​i​
Rewrite 2−1+3​i​ in standard complex form: −21​+23​​i
2−1+3​i​
Apply the fraction rule: ca±b​=ca​±cb​2−1+3​i​=−21​+23​i​=−21​+23​i​
=−21​+23​​i
=−21​+i23​​

Popular Examples

arcsin((8.3)/(16))6sin((4pi)/3)arccot(sqrt(7))cos(90+30)cos(3(pi))

Frequently Asked Questions (FAQ)

  • What is the value of (cos(pi/3)+isin(pi/3))^2 ?

    The value of (cos(pi/3)+isin(pi/3))^2 is -1/2+i(sqrt(3))/2
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