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Popular Trigonometry >

2sec^2(pi/6)tan(pi/6)

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Solution

2sec2(6π​)tan(6π​)

Solution

983​​
+1
Decimal
1.53960…
Solution steps
2sec2(6π​)tan(6π​)
Use the following trivial identity:sec(6π​)=323​​
sec(6π​)
sec(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​sec(x)1323​​2​2Undefined−2−2​−323​​​xπ67π​45π​34π​23π​35π​47π​611π​​sec(x)−1−323​​−2​−2Undefined22​323​​​​
=323​​
Use the following trivial identity:tan(6π​)=33​​
tan(6π​)
tan(x) periodicity table with πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​tan(x)033​​13​±∞−3​−1−33​​​​
=33​​
=2(323​​)233​​
Simplify 2(323​​)233​​:983​​
2(323​​)233​​
(323​​)2=322​
(323​​)2
Apply exponent rule: (ba​)c=bcac​=32(23​)2​
Apply exponent rule: (a⋅b)n=anbn(23​)2=22(3​)2=3222(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
=3222⋅3​
Cancel the common factor: 3=322​
=2⋅322​⋅33​​
Multiply fractions: a⋅cb​⋅ed​=c⋅ea⋅b⋅d​=3⋅3223​⋅2​
223​⋅2=233​
223​⋅2
Apply exponent rule: ab⋅ac=ab+c22⋅2=22+1=3​⋅22+1
Add the numbers: 2+1=3=3​⋅23
=3⋅3233​​
Multiply the numbers: 3⋅3=9=9233​​
Factor 9:32
Factor 9=32
=32233​​
Cancel 323​⋅23​:323​23​
323​⋅23​
Apply radical rule: 3​=321​=3223⋅321​​
Apply exponent rule: xbxa​=xb−a1​32321​​=32−21​1​=32−21​23​
Subtract the numbers: 2−21​=23​=323​23​
=323​23​
323​=33​
323​
323​=31+21​=31+21​
Apply exponent rule: xa+b=xaxb=31⋅321​
Refine=33​
=33​23​
23=8=33​8​
Rationalize 33​8​:983​​
33​8​
Multiply by the conjugate 3​3​​=33​3​83​​
33​3​=9
33​3​
Apply exponent rule: ab⋅ac=ab+c33​3​=3⋅321​⋅321​=31+21​+21​=31+21​+21​
Add similar elements: 21​+21​=2⋅21​=31+2⋅21​
2⋅21​=1
2⋅21​
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=31+1
Add the numbers: 1+1=2=32
32=9=9
=983​​
=983​​
=983​​

Popular Examples

csc((7pi)/(12))2cos(-pi)(cos(30))/(cos(45))arctan((-8)/6)cos(30)+sin(45)

Frequently Asked Questions (FAQ)

  • What is the value of 2sec^2(pi/6)tan(pi/6) ?

    The value of 2sec^2(pi/6)tan(pi/6) is (8sqrt(3))/9
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