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Popular Trigonometry >

10picos(2pi*0.8)+e^{3*0.8}*3

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Solution

10πcos(2π⋅0.8)+e3⋅0.8⋅3

Solution

10π(1−45−5​​)+3e512​
+1
Decimal
42.77758…
Solution steps
10πcos(2π0.8)+e3⋅0.8⋅3
=10πcos(2π54​)+e3⋅54​⋅3
Rewrite using trig identities:cos(2π54​)=1−2sin2(54π​)
cos(2π54​)
Use the Double Angle identity: cos(2x)=1−2sin2(x)=1−2sin2(π54​)
Simplify:π54​=54π​
π54​
Multiply fractions: a⋅cb​=ca⋅b​=54π​
=1−2sin2(54π​)
=10π(1−2sin2(54π​))+e3⋅54​⋅3
10π(1−2sin2(54π​))+e3⋅54​⋅3=10π(1−2sin2(54π​))+3e512​
10π(1−2sin2(54π​))+e3⋅54​⋅3
e3⋅54​=e512​
e3⋅54​
Multiply 3⋅54​:512​
3⋅54​
Multiply fractions: a⋅cb​=ca⋅b​=54⋅3​
Multiply the numbers: 4⋅3=12=512​
=e512​
=10π(−2sin2(54π​)+1)+3e512​
=10π(1−2sin2(54π​))+3e512​
Rewrite using trig identities:sin(54π​)=42​5−5​​​
sin(54π​)
Rewrite using trig identities:sin(5π​)
sin(54π​)
Use the basic trigonometric identity: sin(x)=sin(π−x)=sin(π−54π​)
Simplify:π−54π​=5π​
π−54π​
Convert element to fraction: π=5π5​=5π5​−54π​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=5π5−4π​
Add similar elements: 5π−4π=π=5π​
=sin(5π​)
=sin(5π​)
Rewrite using trig identities:42​5−5​​​
sin(5π​)
Show that: cos(5π​)−sin(10π​)=21​
Use the following product to sum identity: 2sin(x)cos(y)=sin(x+y)−sin(x−y)2cos(5π​)sin(10π​)=sin(103π​)−sin(10π​)
Show that: 2cos(5π​)sin(10π​)=21​
Use the Double Angle identity: sin(2x)=2sin(x)cos(x)sin(52π​)=2sin(5π​)cos(5π​)sin(52π​)sin(5π​)=4sin(5π​)sin(10π​)cos(5π​)cos(10π​)
Divide both sides by sin(5π​)sin(52π​)=4sin(10π​)cos(5π​)cos(10π​)
Use the following identity: sin(x)=cos(2π​−x)sin(52π​)=cos(2π​−52π​)cos(2π​−52π​)=4sin(10π​)cos(5π​)cos(10π​)
cos(10π​)=4sin(10π​)cos(5π​)cos(10π​)
Divide both sides by cos(10π​)1=4sin(10π​)cos(5π​)
Divide both sides by 221​=2sin(10π​)cos(5π​)
Substitute 21​=2sin(10π​)cos(5π​)21​=sin(103π​)−sin(10π​)
sin(103π​)=cos(2π​−103π​)21​=cos(2π​−103π​)−sin(10π​)
21​=cos(5π​)−sin(10π​)
Show that: cos(5π​)+sin(10π​)=45​​
Use the factorization rule: a2−b2=(a+b)(a−b)a=cos(5π​)+sin(10π​)(cos(5π​)+sin(10π​))2−(cos(5π​)−sin(10π​))2=((cos(5π​)+sin(10π​))+(cos(5π​)−sin(10π​)))((cos(5π​)+sin(10π​))−(cos(5π​)−sin(10π​)))
Refine(cos(5π​)+sin(10π​))2−(cos(5π​)−sin(10π​))2=2(2cos(5π​)sin(10π​))
Show that: 2cos(5π​)sin(10π​)=21​
Use the Double Angle identity: sin(2x)=2sin(x)cos(x)sin(52π​)=2sin(5π​)cos(5π​)sin(52π​)sin(5π​)=4sin(5π​)sin(10π​)cos(5π​)cos(10π​)
Divide both sides by sin(5π​)sin(52π​)=4sin(10π​)cos(5π​)cos(10π​)
Use the following identity: sin(x)=cos(2π​−x)sin(52π​)=cos(2π​−52π​)cos(2π​−52π​)=4sin(10π​)cos(5π​)cos(10π​)
cos(10π​)=4sin(10π​)cos(5π​)cos(10π​)
Divide both sides by cos(10π​)1=4sin(10π​)cos(5π​)
Divide both sides by 221​=2sin(10π​)cos(5π​)
Substitute 2cos(5π​)sin(10π​)=21​(cos(5π​)+sin(10π​))2−(cos(5π​)−sin(10π​))2=1
Substitute cos(5π​)−sin(10π​)=21​(cos(5π​)+sin(10π​))2−(21​)2=1
Refine(cos(5π​)+sin(10π​))2−41​=1
Add 41​ to both sides(cos(5π​)+sin(10π​))2−41​+41​=1+41​
Refine(cos(5π​)+sin(10π​))2=45​
Take the square root of both sidescos(5π​)+sin(10π​)=±45​​
cos(5π​)cannot be negativesin(10π​)cannot be negativecos(5π​)+sin(10π​)=45​​
Add the following equationscos(5π​)+sin(10π​)=25​​((cos(5π​)+sin(10π​))+(cos(5π​)−sin(10π​)))=(25​​+21​)
Refinecos(5π​)=45​+1​
Square both sides(cos(5π​))2=(45​+1​)2
Use the following identity: sin2(x)=1−cos2(x)sin2(5π​)=1−cos2(5π​)
Substitute cos(5π​)=45​+1​sin2(5π​)=1−(45​+1​)2
Refinesin2(5π​)=85−5​​
Take the square root of both sidessin(5π​)=±85−5​​​
sin(5π​)cannot be negativesin(5π​)=85−5​​​
Refinesin(5π​)=225−5​​​​
=225−5​​​​
225−5​​​​=42​5−5​​​
225−5​​​​
25−5​​​=2​5−5​​​
25−5​​​
Apply radical rule: nba​​=nb​na​​, assuming a≥0,b≥0=2​5−5​​​
=22​5−5​​​​
Apply the fraction rule: acb​​=c⋅ab​=2​⋅25−5​​​
Rationalize 22​5−5​​​:42​5−5​​​
22​5−5​​​
Multiply by the conjugate 2​2​​=2​⋅22​5−5​​2​​
2​⋅22​=4
2​⋅22​
Apply exponent rule: ab⋅ac=ab+c22​2​=2⋅221​⋅221​=21+21​+21​=21+21​+21​
Add similar elements: 21​+21​=2⋅21​=21+2⋅21​
2⋅21​=1
2⋅21​
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=21+1
Add the numbers: 1+1=2=22
22=4=4
=42​5−5​​​
=42​5−5​​​
=42​5−5​​​
=42​5−5​​​
=10π​1−2(42​5−5​​​)2​+3e512​
Simplify 10π​1−2(42​5−5​​​)2​+3e512​:10π(1−45−5​​)+3e512​
10π​1−2(42​5−5​​​)2​+3e512​
2(42​5−5​​​)2=45−5​​
2(42​5−5​​​)2
(42​5−5​​​)2=235−5​​
(42​5−5​​​)2
Apply exponent rule: (ba​)c=bcac​=42(2​5−5​​)2​
Apply exponent rule: (a⋅b)n=anbn(2​5−5​​)2=(2​)2(5−5​​)2=42(2​)2(5−5​​)2​
(2​)2:2
Apply radical rule: a​=a21​=(221​)2
Apply exponent rule: (ab)c=abc=221​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=2
=422(5−5​​)2​
(5−5​​)2:5−5​
Apply radical rule: a​=a21​=((5−5​)21​)2
Apply exponent rule: (ab)c=abc=(5−5​)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=5−5​
=422(5−5​)​
Factor 42:24
Factor 4=22=(22)2
Simplify (22)2:24
(22)2
Apply exponent rule: (ab)c=abc=22⋅2
Multiply the numbers: 2⋅2=4=24
=24
=242(5−5​)​
Cancel the common factor: 2=235−5​​
=2⋅235−5​​
Multiply fractions: a⋅cb​=ca⋅b​=23(5−5​)⋅2​
Cancel the common factor: 2=225−5​​
22=4=45−5​​
=10π(−45−5​​+1)+3e512​
=10π(1−45−5​​)+3e512​

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Frequently Asked Questions (FAQ)

  • What is the value of 10picos(2pi*0.8)+e^{3*0.8}*3 ?

    The value of 10picos(2pi*0.8)+e^{3*0.8}*3 is 10pi(1-(5-sqrt(5))/4)+3e^{12/5}
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