Trigonometry Examples

Solve for x tan((7pi)/4+0)=(-cos(x)+sin(x))/(cos(x)+sin(x))
Step 1
Rewrite the equation as .
Step 2
Divide each term in the equation by .
Step 3
Separate fractions.
Step 4
Convert from to .
Step 5
Divide by .
Step 6
Combine and .
Step 7
Factor out of .
Step 8
Factor out of .
Step 9
Factor out of .
Step 10
Simplify the expression.
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Step 10.1
Rewrite as .
Step 10.2
Move the negative in front of the fraction.
Step 10.3
Reorder factors in .
Step 11
Rewrite in terms of sines and cosines.
Step 12
Rewrite as a product.
Step 13
Multiply by .
Step 14
Simplify the numerator.
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Step 14.1
Add and .
Step 14.2
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 14.3
The exact value of is .
Step 15
Simplify the denominator.
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Step 15.1
Add and .
Step 15.2
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 15.3
The exact value of is .
Step 16
Combine and .
Step 17
Multiply the numerator by the reciprocal of the denominator.
Step 18
Cancel the common factor of .
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Step 18.1
Move the leading negative in into the numerator.
Step 18.2
Factor out of .
Step 18.3
Factor out of .
Step 18.4
Cancel the common factor.
Step 18.5
Rewrite the expression.
Step 19
Cancel the common factor of .
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Step 19.1
Cancel the common factor.
Step 19.2
Rewrite the expression.
Step 20
Convert from to .
Step 21
Simplify the left side.
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Step 21.1
Simplify .
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Step 21.1.1
Rewrite in terms of sines and cosines.
Step 21.1.2
Factor out of .
Step 21.1.3
Multiply by .
Step 22
Simplify the right side.
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Step 22.1
Rewrite in terms of sines and cosines.
Step 23
Multiply both sides of the equation by .
Step 24
Rewrite using the commutative property of multiplication.
Step 25
Cancel the common factor of .
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Step 25.1
Factor out of .
Step 25.2
Cancel the common factor.
Step 25.3
Rewrite the expression.
Step 26
Rewrite using the commutative property of multiplication.
Step 27
Cancel the common factor of .
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Step 27.1
Factor out of .
Step 27.2
Cancel the common factor.
Step 27.3
Rewrite the expression.
Step 28
Divide each term in the equation by .
Step 29
Multiply the numerator by the reciprocal of the denominator.
Step 30
Convert from to .
Step 31
Combine and .
Step 32
Separate fractions.
Step 33
Convert from to .
Step 34
Divide by .
Step 35
Simplify the left side.
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Step 35.1
Simplify .
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Step 35.1.1
Rewrite in terms of sines and cosines.
Step 35.1.2
Factor out of .
Step 35.1.3
Multiply by .
Step 36
Simplify the right side.
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Step 36.1
Rewrite in terms of sines and cosines.
Step 37
Multiply both sides of the equation by .
Step 38
Rewrite using the commutative property of multiplication.
Step 39
Cancel the common factor of .
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Step 39.1
Factor out of .
Step 39.2
Cancel the common factor.
Step 39.3
Rewrite the expression.
Step 40
Rewrite using the commutative property of multiplication.
Step 41
Cancel the common factor of .
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Step 41.1
Factor out of .
Step 41.2
Cancel the common factor.
Step 41.3
Rewrite the expression.
Step 42
Divide each term in the equation by .
Step 43
Multiply the numerator by the reciprocal of the denominator.
Step 44
Convert from to .
Step 45
Combine and .
Step 46
Separate fractions.
Step 47
Convert from to .
Step 48
Divide by .
Step 49
Simplify the left side.
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Step 49.1
Simplify .
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Step 49.1.1
Rewrite in terms of sines and cosines.
Step 49.1.2
Factor out of .
Step 49.1.3
Multiply by .
Step 50
Simplify the right side.
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Step 50.1
Rewrite in terms of sines and cosines.
Step 51
Multiply both sides of the equation by .
Step 52
Rewrite using the commutative property of multiplication.
Step 53
Cancel the common factor of .
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Step 53.1
Factor out of .
Step 53.2
Cancel the common factor.
Step 53.3
Rewrite the expression.
Step 54
Rewrite using the commutative property of multiplication.
Step 55
Cancel the common factor of .
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Step 55.1
Factor out of .
Step 55.2
Cancel the common factor.
Step 55.3
Rewrite the expression.
Step 56
Divide each term in the equation by .
Step 57
Multiply the numerator by the reciprocal of the denominator.
Step 58
Convert from to .
Step 59
Combine and .
Step 60
Separate fractions.
Step 61
Convert from to .
Step 62
Divide by .
Step 63
Simplify the left side.
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Step 63.1
Simplify .
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Step 63.1.1
Rewrite in terms of sines and cosines.
Step 63.1.2
Factor out of .
Step 63.1.3
Multiply by .
Step 64
Simplify the right side.
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Step 64.1
Rewrite in terms of sines and cosines.
Step 65
Multiply both sides of the equation by .
Step 66
Rewrite using the commutative property of multiplication.
Step 67
Cancel the common factor of .
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Step 67.1
Factor out of .
Step 67.2
Cancel the common factor.
Step 67.3
Rewrite the expression.
Step 68
Rewrite using the commutative property of multiplication.
Step 69
Cancel the common factor of .
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Step 69.1
Factor out of .
Step 69.2
Cancel the common factor.
Step 69.3
Rewrite the expression.
Step 70
Divide each term in the equation by .
Step 71
Multiply the numerator by the reciprocal of the denominator.
Step 72
Convert from to .
Step 73
Combine and .
Step 74
Separate fractions.
Step 75
Convert from to .
Step 76
Divide by .
Step 77
Simplify the left side.
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Step 77.1
Simplify .
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Step 77.1.1
Rewrite in terms of sines and cosines.
Step 77.1.2
Factor out of .
Step 77.1.3
Multiply by .
Step 78
Divide each term in the equation by .
Step 79
Multiply the numerator by the reciprocal of the denominator.
Step 80
Convert from to .
Step 81
Combine and .
Step 82
Separate fractions.
Step 83
Rewrite in terms of sines and cosines.
Step 84
Rewrite as a product.
Step 85
Simplify.
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Step 85.1
Convert from to .
Step 85.2
Convert from to .
Step 85.3
Raise to the power of .
Step 85.4
Raise to the power of .
Step 85.5
Use the power rule to combine exponents.
Step 85.6
Add and .
Step 86
Combine and .
Step 87
Reorder factors in .
Step 88
Separate fractions.
Step 89
Rewrite in terms of sines and cosines.
Step 90
Rewrite as a product.
Step 91
Combine fractions.
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Step 91.1
Multiply by .
Step 91.2
Divide by .
Step 92
Simplify the denominator.
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Step 92.1
Use the power rule to combine exponents.
Step 92.2
Add and .
Step 93
Simplify with factoring out.
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Step 93.1
Rewrite as .
Step 93.2
Rewrite as .
Step 94
Convert from to .
Step 95
Replace the with based on the identity.
Step 96
Multiply both sides by .
Step 97
Simplify.
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Step 97.1
Simplify the left side.
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Step 97.1.1
Simplify .
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Step 97.1.1.1
Cancel the common factor of .
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Step 97.1.1.1.1
Cancel the common factor.
Step 97.1.1.1.2
Rewrite the expression.
Step 97.1.1.2
Rearrange terms.
Step 97.1.1.3
Apply pythagorean identity.
Step 97.1.1.4
Rewrite in terms of sines and cosines.
Step 97.1.1.5
Simplify terms.
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Step 97.1.1.5.1
Apply the product rule to .
Step 97.1.1.5.2
One to any power is one.
Step 97.1.1.5.3
Apply the distributive property.
Step 97.1.1.5.4
Cancel the common factor of .
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Step 97.1.1.5.4.1
Factor out of .
Step 97.1.1.5.4.2
Cancel the common factor.
Step 97.1.1.5.4.3
Rewrite the expression.
Step 97.1.1.5.5
Rewrite using the commutative property of multiplication.
Step 97.1.1.5.6
Combine and .
Step 97.1.1.6
Simplify each term.
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Step 97.1.1.6.1
Convert from to .
Step 97.1.1.6.2
Factor out of .
Step 97.1.1.6.3
Separate fractions.
Step 97.1.1.6.4
Convert from to .
Step 97.1.1.6.5
Convert from to .
Step 97.2
Simplify the right side.
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Step 97.2.1
Simplify .
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Step 97.2.1.1
Rewrite in terms of sines and cosines.
Step 97.2.1.2
Simplify terms.
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Step 97.2.1.2.1
Apply the product rule to .
Step 97.2.1.2.2
One to any power is one.
Step 97.2.1.2.3
Apply the distributive property.
Step 97.2.1.2.4
Cancel the common factor of .
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Step 97.2.1.2.4.1
Move the leading negative in into the numerator.
Step 97.2.1.2.4.2
Factor out of .
Step 97.2.1.2.4.3
Cancel the common factor.
Step 97.2.1.2.4.4
Rewrite the expression.
Step 97.2.1.2.5
Combine and .
Step 97.2.1.2.6
Move the negative in front of the fraction.
Step 97.2.1.3
Simplify each term.
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Step 97.2.1.3.1
Convert from to .
Step 97.2.1.3.2
Factor out of .
Step 97.2.1.3.3
Separate fractions.
Step 97.2.1.3.4
Convert from to .
Step 97.2.1.3.5
Convert from to .
Step 98
Solve for .
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Step 98.1
Simplify the left side.
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Step 98.1.1
Simplify each term.
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Step 98.1.1.1
Rewrite in terms of sines and cosines.
Step 98.1.1.2
Rewrite in terms of sines and cosines.
Step 98.1.1.3
Rewrite in terms of sines and cosines.
Step 98.1.1.4
Multiply .
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Step 98.1.1.4.1
Multiply by .
Step 98.1.1.4.2
Raise to the power of .
Step 98.1.1.4.3
Raise to the power of .
Step 98.1.1.4.4
Use the power rule to combine exponents.
Step 98.1.1.4.5
Add and .
Step 98.2
Simplify the right side.
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Step 98.2.1
Simplify each term.
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Step 98.2.1.1
Rewrite in terms of sines and cosines.
Step 98.2.1.2
Rewrite in terms of sines and cosines.
Step 98.2.1.3
Rewrite in terms of sines and cosines.
Step 98.2.1.4
Multiply .
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Step 98.2.1.4.1
Multiply by .
Step 98.2.1.4.2
Raise to the power of .
Step 98.2.1.4.3
Raise to the power of .
Step 98.2.1.4.4
Use the power rule to combine exponents.
Step 98.2.1.4.5
Add and .
Step 98.3
Multiply both sides of the equation by .
Step 98.4
Apply the distributive property.
Step 98.5
Cancel the common factor of .
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Step 98.5.1
Cancel the common factor.
Step 98.5.2
Rewrite the expression.
Step 98.6
Rewrite using the commutative property of multiplication.
Step 98.7
Cancel the common factor of .
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Step 98.7.1
Factor out of .
Step 98.7.2
Factor out of .
Step 98.7.3
Cancel the common factor.
Step 98.7.4
Rewrite the expression.
Step 98.8
Apply the distributive property.
Step 98.9
Rewrite using the commutative property of multiplication.
Step 98.10
Rewrite using the commutative property of multiplication.
Step 98.11
Simplify each term.
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Step 98.11.1
Cancel the common factor of .
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Step 98.11.1.1
Factor out of .
Step 98.11.1.2
Cancel the common factor.
Step 98.11.1.3
Rewrite the expression.
Step 98.11.2
Cancel the common factor of .
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Step 98.11.2.1
Factor out of .
Step 98.11.2.2
Factor out of .
Step 98.11.2.3
Cancel the common factor.
Step 98.11.2.4
Rewrite the expression.
Step 98.12
Convert from to .
Step 98.13
Move all terms containing to the left side of the equation.
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Step 98.13.1
Add to both sides of the equation.
Step 98.13.2
Add and .
Step 98.14
Since , there are no solutions.
No solution
No solution