Trigonometry Examples

Solve for x 13=3sin((2pi)/365*(x-79))+12
Step 1
Rewrite the equation as .
Step 2
Move all terms not containing to the right side of the equation.
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Step 2.1
Subtract from both sides of the equation.
Step 2.2
Subtract from .
Step 3
Divide each term in by and simplify.
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Step 3.1
Divide 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
Divide by .
Step 4
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5
Simplify the left side.
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Step 5.1
Simplify .
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Step 5.1.1
Apply the distributive property.
Step 5.1.2
Combine and .
Step 5.1.3
Multiply .
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Step 5.1.3.1
Combine and .
Step 5.1.3.2
Multiply by .
Step 5.1.4
Move the negative in front of the fraction.
Step 6
Simplify the right side.
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Step 6.1
Evaluate .
Step 7
Move all terms not containing to the right side of the equation.
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Step 7.1
Add to both sides of the equation.
Step 7.2
Add and .
Step 8
Multiply both sides of the equation by .
Step 9
Simplify both sides of the equation.
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Step 9.1
Simplify the left side.
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Step 9.1.1
Simplify .
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Step 9.1.1.1
Cancel the common factor of .
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Step 9.1.1.1.1
Cancel the common factor.
Step 9.1.1.1.2
Rewrite the expression.
Step 9.1.1.2
Cancel the common factor of .
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Step 9.1.1.2.1
Factor out of .
Step 9.1.1.2.2
Cancel the common factor.
Step 9.1.1.2.3
Rewrite the expression.
Step 9.2
Simplify the right side.
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Step 9.2.1
Simplify .
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Step 9.2.1.1
Multiply .
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Step 9.2.1.1.1
Combine and .
Step 9.2.1.1.2
Multiply by .
Step 9.2.1.2
Replace with an approximation.
Step 9.2.1.3
Multiply by .
Step 9.2.1.4
Divide by .
Step 10
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 11
Solve for .
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Step 11.1
Subtract from .
Step 11.2
Move all terms not containing to the right side of the equation.
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Step 11.2.1
Add to both sides of the equation.
Step 11.2.2
Add and .
Step 11.3
Multiply both sides of the equation by .
Step 11.4
Simplify both sides of the equation.
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Step 11.4.1
Simplify the left side.
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Step 11.4.1.1
Simplify .
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Step 11.4.1.1.1
Cancel the common factor of .
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Step 11.4.1.1.1.1
Cancel the common factor.
Step 11.4.1.1.1.2
Rewrite the expression.
Step 11.4.1.1.2
Cancel the common factor of .
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Step 11.4.1.1.2.1
Factor out of .
Step 11.4.1.1.2.2
Cancel the common factor.
Step 11.4.1.1.2.3
Rewrite the expression.
Step 11.4.2
Simplify the right side.
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Step 11.4.2.1
Simplify .
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Step 11.4.2.1.1
Multiply .
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Step 11.4.2.1.1.1
Combine and .
Step 11.4.2.1.1.2
Multiply by .
Step 11.4.2.1.2
Replace with an approximation.
Step 11.4.2.1.3
Multiply by .
Step 11.4.2.1.4
Divide by .
Step 12
Find the period of .
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Step 12.1
The period of the function can be calculated using .
Step 12.2
Replace with in the formula for period.
Step 12.3
is approximately which is positive so remove the absolute value
Step 12.4
Multiply the numerator by the reciprocal of the denominator.
Step 12.5
Cancel the common factor of .
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Step 12.5.1
Cancel the common factor.
Step 12.5.2
Rewrite the expression.
Step 13
The period of the function is so values will repeat every radians in both directions.
, for any integer