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Popular Trigonometry >

(sin(15)-cos(15))/(tan(15))

  • Pre Algebra
  • Algebra
  • Pre Calculus
  • Calculus
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Solution

tan(15∘)sin(15∘)−cos(15∘)​

Solution

−222​+6​​
+1
Decimal
−2.63895…
Solution steps
tan(15∘)sin(15∘)−cos(15∘)​
Rewrite using trig identities:tan(15∘)=cos(15∘)sin(15∘)​
tan(15∘)
Use the basic trigonometric identity: tan(x)=cos(x)sin(x)​=cos(15∘)sin(15∘)​
=cos(15∘)sin(15∘)​sin(15∘)−cos(15∘)​
Simplify=sin(15∘)cos(15∘)sin(15∘)−cos2(15∘)​
Rewrite using trig identities:cos(15∘)=46​+2​​
cos(15∘)
Rewrite using trig identities:cos(45∘)cos(30∘)+sin(45∘)sin(30∘)
cos(15∘)
Write cos(15∘)as cos(45∘−30∘)=cos(45∘−30∘)
Use the Angle Difference identity: cos(s−t)=cos(s)cos(t)+sin(s)sin(t)=cos(45∘)cos(30∘)+sin(45∘)sin(30∘)
=cos(45∘)cos(30∘)+sin(45∘)sin(30∘)
Use the following trivial identity:cos(45∘)=22​​
cos(45∘)
cos(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​cos(x)123​​22​​21​0−21​−22​​−23​​​x180∘210∘225∘240∘270∘300∘315∘330∘​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
=22​​
Use the following trivial identity:cos(30∘)=23​​
cos(30∘)
cos(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​cos(x)123​​22​​21​0−21​−22​​−23​​​x180∘210∘225∘240∘270∘300∘315∘330∘​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
=23​​
Use the following trivial identity:sin(45∘)=22​​
sin(45∘)
sin(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​sin(x)021​22​​23​​123​​22​​21​​x180∘210∘225∘240∘270∘300∘315∘330∘​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
=22​​
Use the following trivial identity:sin(30∘)=21​
sin(30∘)
sin(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​sin(x)021​22​​23​​123​​22​​21​​x180∘210∘225∘240∘270∘300∘315∘330∘​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
=21​
=22​​⋅23​​+22​​⋅21​
Simplify 22​​⋅23​​+22​​⋅21​:46​+2​​
22​​⋅23​​+22​​⋅21​
22​​⋅23​​=46​​
22​​⋅23​​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅22​3​​
Multiply the numbers: 2⋅2=4=42​3​​
Simplify 2​3​:6​
2​3​
Apply radical rule: a​b​=a⋅b​2​3​=2⋅3​=2⋅3​
Multiply the numbers: 2⋅3=6=6​
=46​​
22​​⋅21​=42​​
22​​⋅21​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅22​⋅1​
Multiply: 2​⋅1=2​=2⋅22​​
Multiply the numbers: 2⋅2=4=42​​
=46​​+42​​
Apply rule ca​±cb​=ca±b​=46​+2​​
=46​+2​​
Rewrite using trig identities:sin(15∘)=46​−2​​
sin(15∘)
Rewrite using trig identities:sin(45∘)cos(30∘)−cos(45∘)sin(30∘)
sin(15∘)
Write sin(15∘)as sin(45∘−30∘)=sin(45∘−30∘)
Use the Angle Difference identity: sin(s−t)=sin(s)cos(t)−cos(s)sin(t)=sin(45∘)cos(30∘)−cos(45∘)sin(30∘)
=sin(45∘)cos(30∘)−cos(45∘)sin(30∘)
Use the following trivial identity:sin(45∘)=22​​
sin(45∘)
sin(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​sin(x)021​22​​23​​123​​22​​21​​x180∘210∘225∘240∘270∘300∘315∘330∘​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
=22​​
Use the following trivial identity:cos(30∘)=23​​
cos(30∘)
cos(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​cos(x)123​​22​​21​0−21​−22​​−23​​​x180∘210∘225∘240∘270∘300∘315∘330∘​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
=23​​
Use the following trivial identity:cos(45∘)=22​​
cos(45∘)
cos(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​cos(x)123​​22​​21​0−21​−22​​−23​​​x180∘210∘225∘240∘270∘300∘315∘330∘​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
=22​​
Use the following trivial identity:sin(30∘)=21​
sin(30∘)
sin(x) periodicity table with 360∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​sin(x)021​22​​23​​123​​22​​21​​x180∘210∘225∘240∘270∘300∘315∘330∘​sin(x)0−21​−22​​−23​​−1−23​​−22​​−21​​​
=21​
=22​​⋅23​​−22​​⋅21​
Simplify 22​​⋅23​​−22​​⋅21​:46​−2​​
22​​⋅23​​−22​​⋅21​
22​​⋅23​​=46​​
22​​⋅23​​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅22​3​​
Multiply the numbers: 2⋅2=4=42​3​​
Simplify 2​3​:6​
2​3​
Apply radical rule: a​b​=a⋅b​2​3​=2⋅3​=2⋅3​
Multiply the numbers: 2⋅3=6=6​
=46​​
22​​⋅21​=42​​
22​​⋅21​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=2⋅22​⋅1​
Multiply: 2​⋅1=2​=2⋅22​​
Multiply the numbers: 2⋅2=4=42​​
=46​​−42​​
Apply rule ca​±cb​=ca±b​=46​−2​​
=46​−2​​
=46​−2​​46​+2​​⋅46​−2​​−(46​+2​​)2​
Simplify 46​−2​​46​+2​​⋅46​−2​​−(46​+2​​)2​:−222​+6​​
46​−2​​46​+2​​⋅46​−2​​−(46​+2​​)2​
Apply the fraction rule: cb​a​=ba⋅c​=6​−2​(46​+2​​⋅46​−2​​−(46​+2​​)2)⋅4​
46​+2​​⋅46​−2​​=41​
46​+2​​⋅46​−2​​
Multiply fractions: ba​⋅dc​=b⋅da⋅c​=4⋅4(6​+2​)(6​−2​)​
Multiply the numbers: 4⋅4=16=16(6​+2​)(6​−2​)​
Expand (6​+2​)(6​−2​):4
(6​+2​)(6​−2​)
Apply Difference of Two Squares Formula: (a+b)(a−b)=a2−b2a=6​,b=2​=(6​)2−(2​)2
Simplify (6​)2−(2​)2:4
(6​)2−(2​)2
(6​)2=6
(6​)2
Apply radical rule: a​=a21​=(621​)2
Apply exponent rule: (ab)c=abc=621​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=6
(2​)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
=6−2
Subtract the numbers: 6−2=4=4
=4
=164​
Cancel the common factor: 4=41​
(46​+2​​)2=42+3​​
(46​+2​​)2
Apply exponent rule: (ba​)c=bcac​=42(6​+2​)2​
(6​+2​)2=8+43​
(6​+2​)2
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2a=6​,b=2​
=(6​)2+26​2​+(2​)2
Simplify (6​)2+26​2​+(2​)2:8+43​
(6​)2+26​2​+(2​)2
(6​)2=6
(6​)2
Apply radical rule: a​=a21​=(621​)2
Apply exponent rule: (ab)c=abc=621​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=6
26​2​=43​
26​2​
Factor integer 6=2⋅3=22⋅3​2​
Apply radical rule: nab​=na​nb​2⋅3​=2​3​=22​3​2​
Apply radical rule: a​a​=a2​2​=2=2⋅23​
Multiply the numbers: 2⋅2=4=43​
(2​)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
=6+43​+2
Add the numbers: 6+2=8=8+43​
=8+43​
=428+43​​
Factor 8+43​:4(2+3​)
8+43​
Rewrite as=4⋅2+43​
Factor out common term 4=4(2+3​)
=424(2+3​)​
Cancel the common factor: 4=42+3​​
=6​−2​4(41​−42+3​​)​
Combine the fractions 41​−42+3​​:41−(2+3​)​
Apply rule ca​±cb​=ca±b​=41−(2+3​)​
=6​−2​4(4−(2+3​)+1​)​
Remove parentheses: (a)=a=6​−2​41−(2+3​)​⋅4​
Multiply 41−(2+3​)​⋅4:−1−3​
41−(2+3​)​⋅4
Multiply fractions: a⋅cb​=ca⋅b​=4(1−(2+3​))⋅4​
Cancel the common factor: 4=1−(2+3​)
−(2+3​):−2−3​
−(2+3​)
Distribute parentheses=−(2)−(3​)
Apply minus-plus rules+(−a)=−a=−2−3​
=1−2−3​
Subtract the numbers: 1−2=−1=−1−3​
=6​−2​−1−3​​
Rationalize 6​−2​−1−3​​:−222​+6​​
6​−2​−1−3​​
Multiply by the conjugate 6​+2​6​+2​​=(6​−2​)(6​+2​)(−1−3​)(6​+2​)​
(−1−3​)(6​+2​)=−26​−42​
(−1−3​)(6​+2​)
Apply FOIL method: (a+b)(c+d)=ac+ad+bc+bda=−1,b=−3​,c=6​,d=2​=(−1)6​+(−1)2​+(−3​)6​+(−3​)2​
Apply minus-plus rules+(−a)=−a=−1⋅6​−1⋅2​−3​6​−3​2​
Simplify −1⋅6​−1⋅2​−3​6​−3​2​:−26​−42​
−1⋅6​−1⋅2​−3​6​−3​2​
1⋅6​=6​
1⋅6​
Multiply: 1⋅6​=6​=6​
1⋅2​=2​
1⋅2​
Multiply: 1⋅2​=2​=2​
3​6​=32​
3​6​
Factor integer 6=3⋅2=3​3⋅2​
Apply radical rule: nab​=na​nb​3⋅2​=3​2​=3​3​2​
Apply radical rule: a​a​=a3​3​=3=32​
3​2​=6​
3​2​
Apply radical rule: a​b​=a⋅b​3​2​=3⋅2​=3⋅2​
Multiply the numbers: 3⋅2=6=6​
=−6​−2​−32​−6​
Add similar elements: −2​−32​=−42​=−6​−42​−6​
Add similar elements: −6​−6​=−26​=−26​−42​
=−26​−42​
(6​−2​)(6​+2​)=4
(6​−2​)(6​+2​)
Apply Difference of Two Squares Formula: (a−b)(a+b)=a2−b2a=6​,b=2​=(6​)2−(2​)2
Simplify (6​)2−(2​)2:4
(6​)2−(2​)2
(6​)2=6
(6​)2
Apply radical rule: a​=a21​=(621​)2
Apply exponent rule: (ab)c=abc=621​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=6
(2​)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
=6−2
Subtract the numbers: 6−2=4=4
=4
=4−26​−42​​
Factor −26​−42​:−2(6​+22​)
−26​−42​
Rewrite as=−26​−2⋅22​
Factor out common term 2=−2(6​+22​)
=−42(6​+22​)​
Cancel the common factor: 2=−222​+6​​
=−222​+6​​
=−222​+6​​

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Frequently Asked Questions (FAQ)

  • What is the value of (sin(15)-cos(15))/(tan(15)) ?

    The value of (sin(15)-cos(15))/(tan(15)) is -(2sqrt(2)+sqrt(6))/2
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