Trigonometry Examples

Solve for x 9/(sin(A))=7/(sin(15))
Step 1
Factor each term.
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Step 1.1
Separate fractions.
Step 1.2
Convert from to .
Step 1.3
Divide by .
Step 1.4
The exact value of is .
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Step 1.4.1
Split into two angles where the values of the six trigonometric functions are known.
Step 1.4.2
Separate negation.
Step 1.4.3
Apply the difference of angles identity.
Step 1.4.4
The exact value of is .
Step 1.4.5
The exact value of is .
Step 1.4.6
The exact value of is .
Step 1.4.7
The exact value of is .
Step 1.4.8
The exact value of is .
Step 1.4.9
The exact value of is .
Step 1.4.10
The exact value of is .
Step 1.4.11
The exact value of is .
Step 1.4.12
Simplify .
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Step 1.4.12.1
Simplify the numerator.
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Step 1.4.12.1.1
Multiply by .
Step 1.4.12.1.2
Combine and .
Step 1.4.12.1.3
Combine and .
Step 1.4.12.2
Simplify the denominator.
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Step 1.4.12.2.1
Multiply by .
Step 1.4.12.2.2
Combine and simplify the denominator.
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Step 1.4.12.2.2.1
Multiply by .
Step 1.4.12.2.2.2
Raise to the power of .
Step 1.4.12.2.2.3
Raise to the power of .
Step 1.4.12.2.2.4
Use the power rule to combine exponents.
Step 1.4.12.2.2.5
Add and .
Step 1.4.12.2.2.6
Rewrite as .
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Step 1.4.12.2.2.6.1
Use to rewrite as .
Step 1.4.12.2.2.6.2
Apply the power rule and multiply exponents, .
Step 1.4.12.2.2.6.3
Combine and .
Step 1.4.12.2.2.6.4
Cancel the common factor of .
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Step 1.4.12.2.2.6.4.1
Cancel the common factor.
Step 1.4.12.2.2.6.4.2
Rewrite the expression.
Step 1.4.12.2.2.6.5
Evaluate the exponent.
Step 1.4.12.2.3
Cancel the common factor of .
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Step 1.4.12.2.3.1
Cancel the common factor.
Step 1.4.12.2.3.2
Rewrite the expression.
Step 1.4.12.2.4
Multiply by .
Step 1.4.12.2.5
Combine and simplify the denominator.
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Step 1.4.12.2.5.1
Multiply by .
Step 1.4.12.2.5.2
Raise to the power of .
Step 1.4.12.2.5.3
Raise to the power of .
Step 1.4.12.2.5.4
Use the power rule to combine exponents.
Step 1.4.12.2.5.5
Add and .
Step 1.4.12.2.5.6
Rewrite as .
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Step 1.4.12.2.5.6.1
Use to rewrite as .
Step 1.4.12.2.5.6.2
Apply the power rule and multiply exponents, .
Step 1.4.12.2.5.6.3
Combine and .
Step 1.4.12.2.5.6.4
Cancel the common factor of .
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Step 1.4.12.2.5.6.4.1
Cancel the common factor.
Step 1.4.12.2.5.6.4.2
Rewrite the expression.
Step 1.4.12.2.5.6.5
Evaluate the exponent.
Step 1.4.12.2.6
Multiply .
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Step 1.4.12.2.6.1
Combine and .
Step 1.4.12.2.6.2
Combine using the product rule for radicals.
Step 1.4.12.2.6.3
Multiply by .
Step 1.4.12.2.7
To write as a fraction with a common denominator, multiply by .
Step 1.4.12.2.8
Combine and .
Step 1.4.12.2.9
Combine the numerators over the common denominator.
Step 1.4.12.2.10
Multiply by .
Step 1.4.12.3
Simplify the numerator.
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Step 1.4.12.3.1
Multiply by .
Step 1.4.12.3.2
Multiply by .
Step 1.4.12.4
Simplify the denominator.
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Step 1.4.12.4.1
Combine using the product rule for radicals.
Step 1.4.12.4.2
Multiply by .
Step 1.4.12.5
Simplify the numerator.
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Step 1.4.12.5.1
Combine and into a single radical.
Step 1.4.12.5.2
Cancel the common factor of and .
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Step 1.4.12.5.2.1
Factor out of .
Step 1.4.12.5.2.2
Cancel the common factors.
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Step 1.4.12.5.2.2.1
Factor out of .
Step 1.4.12.5.2.2.2
Cancel the common factor.
Step 1.4.12.5.2.2.3
Rewrite the expression.
Step 1.4.12.5.3
Rewrite as .
Step 1.4.12.5.4
Any root of is .
Step 1.4.12.5.5
Multiply by .
Step 1.4.12.5.6
Combine and simplify the denominator.
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Step 1.4.12.5.6.1
Multiply by .
Step 1.4.12.5.6.2
Raise to the power of .
Step 1.4.12.5.6.3
Raise to the power of .
Step 1.4.12.5.6.4
Use the power rule to combine exponents.
Step 1.4.12.5.6.5
Add and .
Step 1.4.12.5.6.6
Rewrite as .
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Step 1.4.12.5.6.6.1
Use to rewrite as .
Step 1.4.12.5.6.6.2
Apply the power rule and multiply exponents, .
Step 1.4.12.5.6.6.3
Combine and .
Step 1.4.12.5.6.6.4
Cancel the common factor of .
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Step 1.4.12.5.6.6.4.1
Cancel the common factor.
Step 1.4.12.5.6.6.4.2
Rewrite the expression.
Step 1.4.12.5.6.6.5
Evaluate the exponent.
Step 1.4.12.5.7
Combine and .
Step 1.4.12.6
Multiply the numerator by the reciprocal of the denominator.
Step 1.4.12.7
Cancel the common factor of .
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Step 1.4.12.7.1
Cancel the common factor.
Step 1.4.12.7.2
Rewrite the expression.
Step 1.4.12.8
Combine and .
Step 1.4.12.9
Combine and .
Step 1.4.12.10
Cancel the common factor of and .
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Step 1.4.12.10.1
Factor out of .
Step 1.4.12.10.2
Cancel the common factors.
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Step 1.4.12.10.2.1
Factor out of .
Step 1.4.12.10.2.2
Factor out of .
Step 1.4.12.10.2.3
Factor out of .
Step 1.4.12.10.2.4
Cancel the common factor.
Step 1.4.12.10.2.5
Rewrite the expression.
Step 1.4.12.11
Multiply by .
Step 1.4.12.12
Multiply by .
Step 1.4.12.13
Expand the denominator using the FOIL method.
Step 1.4.12.14
Simplify.
Step 1.4.12.15
Cancel the common factor of and .
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Step 1.4.12.15.1
Factor out of .
Step 1.4.12.15.2
Cancel the common factors.
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Step 1.4.12.15.2.1
Factor out of .
Step 1.4.12.15.2.2
Cancel the common factor.
Step 1.4.12.15.2.3
Rewrite the expression.
Step 1.4.12.16
Apply the distributive property.
Step 1.4.12.17
Multiply .
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Step 1.4.12.17.1
Combine using the product rule for radicals.
Step 1.4.12.17.2
Multiply by .
Step 1.4.12.18
Combine using the product rule for radicals.
Step 1.4.12.19
Simplify each term.
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Step 1.4.12.19.1
Multiply by .
Step 1.4.12.19.2
Rewrite as .
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Step 1.4.12.19.2.1
Factor out of .
Step 1.4.12.19.2.2
Rewrite as .
Step 1.4.12.19.3
Pull terms out from under the radical.
Step 1.4.12.20
Cancel the common factor of and .
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Step 1.4.12.20.1
Factor out of .
Step 1.4.12.20.2
Factor out of .
Step 1.4.12.20.3
Factor out of .
Step 1.4.12.20.4
Cancel the common factors.
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Step 1.4.12.20.4.1
Factor out of .
Step 1.4.12.20.4.2
Cancel the common factor.
Step 1.4.12.20.4.3
Rewrite the expression.
Step 1.4.12.20.4.4
Divide by .
Step 1.5
Apply the distributive property.
Step 2
Find the LCD of the terms in the equation.
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Step 2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.2
The LCM of one and any expression is the expression.
Step 3
Multiply each term in by to eliminate the fractions.
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Step 3.1
Multiply each term in by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Cancel the common factor of .
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Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Rewrite the expression.
Step 4
Solve the equation.
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Step 4.1
Rewrite the equation as .
Step 4.2
Factor out of .
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Step 4.2.1
Factor out of .
Step 4.2.2
Factor out of .
Step 4.2.3
Factor out of .
Step 4.3
Divide each term in by and simplify.
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Step 4.3.1
Divide each term in by .
Step 4.3.2
Simplify the left side.
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Step 4.3.2.1
Cancel the common factor of and .
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Step 4.3.2.1.1
Factor out of .
Step 4.3.2.1.2
Cancel the common factors.
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Step 4.3.2.1.2.1
Factor out of .
Step 4.3.2.1.2.2
Factor out of .
Step 4.3.2.1.2.3
Factor out of .
Step 4.3.2.1.2.4
Cancel the common factor.
Step 4.3.2.1.2.5
Rewrite the expression.
Step 4.3.2.2
Cancel the common factor of .
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Step 4.3.2.2.1
Cancel the common factor.
Step 4.3.2.2.2
Divide by .
Step 4.3.3
Simplify the right side.
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Step 4.3.3.1
Multiply by .
Step 4.3.3.2
Multiply by .
Step 4.3.3.3
Expand the denominator using the FOIL method.
Step 4.3.3.4
Simplify.
Step 4.3.3.5
Cancel the common factor of and .
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Step 4.3.3.5.1
Factor out of .
Step 4.3.3.5.2
Cancel the common factors.
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Step 4.3.3.5.2.1
Factor out of .
Step 4.3.3.5.2.2
Cancel the common factor.
Step 4.3.3.5.2.3
Rewrite the expression.
Step 5
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6
Simplify the right side.
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Step 6.1
Evaluate .
Step 7
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 8
Subtract from .
Step 9
Find the period of .
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Step 9.1
The period of the function can be calculated using .
Step 9.2
Replace with in the formula for period.
Step 9.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9.4
Divide by .
Step 10
The period of the function is so values will repeat every degrees in both directions.
, for any integer