Trigonometry Examples

Expand Using Sum/Difference Formulas tan(105)
tan(105)tan(105)
Step 1
First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, 105105 can be split into 45+6045+60.
tan(45+60)tan(45+60)
Step 2
Use the sum formula for tangent to simplify the expression. The formula states that tan(A+B)=tan(A)+tan(B)1-tan(A)tan(B)tan(A+B)=tan(A)+tan(B)1tan(A)tan(B).
tan(45)+tan(60)1-tan(45)tan(60)tan(45)+tan(60)1tan(45)tan(60)
Step 3
Simplify the numerator.
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Step 3.1
The exact value of tan(45)tan(45) is 11.
1+tan(60)1-tan(45)tan(60)1+tan(60)1tan(45)tan(60)
Step 3.2
The exact value of tan(60)tan(60) is 33.
1+31-tan(45)tan(60)1+31tan(45)tan(60)
1+31-tan(45)tan(60)1+31tan(45)tan(60)
Step 4
Simplify the denominator.
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Step 4.1
The exact value of tan(45)tan(45) is 11.
1+31-11tan(60)1+3111tan(60)
Step 4.2
Multiply -11 by 11.
1+31-1tan(60)1+311tan(60)
Step 4.3
The exact value of tan(60)tan(60) is 33.
1+31-131+3113
Step 4.4
Rewrite -1313 as -33.
1+31-31+313
1+31-31+313
Step 5
Multiply 1+31-31+313 by 1+31+31+31+3.
1+31-31+31+31+3131+31+3
Step 6
Combine fractions.
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Step 6.1
Multiply 1+31-31+313 by 1+31+31+31+3.
(1+3)(1+3)(1-3)(1+3)(1+3)(1+3)(13)(1+3)
Step 6.2
Expand the denominator using the FOIL method.
(1+3)(1+3)1+3-3-32(1+3)(1+3)1+3332
Step 6.3
Simplify.
(1+3)(1+3)-2(1+3)(1+3)2
(1+3)(1+3)-2(1+3)(1+3)2
Step 7
Simplify the numerator.
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Step 7.1
Raise 1+31+3 to the power of 11.
(1+3)1(1+3)-2(1+3)1(1+3)2
Step 7.2
Raise 1+31+3 to the power of 11.
(1+3)1(1+3)1-2(1+3)1(1+3)12
Step 7.3
Use the power rule aman=am+naman=am+n to combine exponents.
(1+3)1+1-2(1+3)1+12
Step 7.4
Add 11 and 11.
(1+3)2-2(1+3)22
(1+3)2-2(1+3)22
Step 8
Rewrite (1+3)2(1+3)2 as (1+3)(1+3)(1+3)(1+3).
(1+3)(1+3)-2(1+3)(1+3)2
Step 9
Expand (1+3)(1+3)(1+3)(1+3) using the FOIL Method.
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Step 9.1
Apply the distributive property.
1(1+3)+3(1+3)-21(1+3)+3(1+3)2
Step 9.2
Apply the distributive property.
11+13+3(1+3)-211+13+3(1+3)2
Step 9.3
Apply the distributive property.
11+13+31+33-211+13+31+332
11+13+31+33-211+13+31+332
Step 10
Simplify and combine like terms.
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Step 10.1
Simplify each term.
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Step 10.1.1
Multiply 1 by 1.
1+13+31+33-2
Step 10.1.2
Multiply 3 by 1.
1+3+31+33-2
Step 10.1.3
Multiply 3 by 1.
1+3+3+33-2
Step 10.1.4
Combine using the product rule for radicals.
1+3+3+33-2
Step 10.1.5
Multiply 3 by 3.
1+3+3+9-2
Step 10.1.6
Rewrite 9 as 32.
1+3+3+32-2
Step 10.1.7
Pull terms out from under the radical, assuming positive real numbers.
1+3+3+3-2
1+3+3+3-2
Step 10.2
Add 1 and 3.
4+3+3-2
Step 10.3
Add 3 and 3.
4+23-2
4+23-2
Step 11
Cancel the common factor of 4+23 and -2.
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Step 11.1
Factor 2 out of 4.
2(2)+23-2
Step 11.2
Factor 2 out of 23.
2(2)+2(3)-2
Step 11.3
Factor 2 out of 2(2)+2(3).
2(2+3)-2
Step 11.4
Move the negative one from the denominator of 2+3-1.
-1(2+3)
-1(2+3)
Step 12
Rewrite -1(2+3) as -(2+3).
-(2+3)
Step 13
Apply the distributive property.
-12-3
Step 14
Multiply -1 by 2.
-2-3
Step 15
The result can be shown in multiple forms.
Exact Form:
-2-3
Decimal Form:
-3.73205080
 [x2  12  π  xdx ]