Trigonometry Examples

Solve for x sin(2x+pi/6)=1/2
Step 1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2
Simplify the right side.
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Step 2.1
The exact value of is .
Step 3
Move all terms not containing to the right side of the equation.
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Step 3.1
Subtract from both sides of the equation.
Step 3.2
Combine the numerators over the common denominator.
Step 3.3
Subtract from .
Step 3.4
Divide by .
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Divide by .
Step 5
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 6
Solve for .
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Step 6.1
Simplify .
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Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine fractions.
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Step 6.1.2.1
Combine and .
Step 6.1.2.2
Combine the numerators over the common denominator.
Step 6.1.3
Simplify the numerator.
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Step 6.1.3.1
Move to the left of .
Step 6.1.3.2
Subtract from .
Step 6.2
Move all terms not containing to the right side of the equation.
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Step 6.2.1
Subtract from both sides of the equation.
Step 6.2.2
Combine the numerators over the common denominator.
Step 6.2.3
Subtract from .
Step 6.2.4
Cancel the common factor of and .
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Step 6.2.4.1
Factor out of .
Step 6.2.4.2
Cancel the common factors.
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Step 6.2.4.2.1
Factor out of .
Step 6.2.4.2.2
Cancel the common factor.
Step 6.2.4.2.3
Rewrite the expression.
Step 6.3
Divide each term in by and simplify.
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Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
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Step 6.3.2.1
Cancel the common factor of .
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Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.3
Simplify the right side.
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Step 6.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.3.2
Cancel the common factor of .
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Step 6.3.3.2.1
Factor out of .
Step 6.3.3.2.2
Cancel the common factor.
Step 6.3.3.2.3
Rewrite the expression.
Step 7
Find the period of .
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Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Cancel the common factor of .
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Step 7.4.1
Cancel the common factor.
Step 7.4.2
Divide by .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer