Trigonometry Examples

Solve for t 14.55=2.35sin(0.017t+1.86)+12.2
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 3.3
Simplify the right side.
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Step 3.3.1
Divide by .
Step 4
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5
Simplify the right side.
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Step 5.1
The exact value of is .
Step 6
Move all terms not containing to the right side of the equation.
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Step 6.1
Subtract from both sides of the equation.
Step 6.2
Subtract from .
Step 7
Divide each term in by and simplify.
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Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
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Step 7.2.1
Cancel the common factor of .
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Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.3
Simplify the right side.
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Step 7.3.1
Divide by .
Step 8
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 9
Solve for .
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Step 9.1
Simplify .
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Step 9.1.1
To write as a fraction with a common denominator, multiply by .
Step 9.1.2
Combine fractions.
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Step 9.1.2.1
Combine and .
Step 9.1.2.2
Combine the numerators over the common denominator.
Step 9.1.3
Simplify the numerator.
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Step 9.1.3.1
Move to the left of .
Step 9.1.3.2
Subtract from .
Step 9.2
Move all terms not containing to the right side of the equation.
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Step 9.2.1
Subtract from both sides of the equation.
Step 9.2.2
Subtract from .
Step 9.3
Divide each term in by and simplify.
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Step 9.3.1
Divide each term in by .
Step 9.3.2
Simplify the left side.
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Step 9.3.2.1
Cancel the common factor of .
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Step 9.3.2.1.1
Cancel the common factor.
Step 9.3.2.1.2
Divide by .
Step 9.3.3
Simplify the right side.
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Step 9.3.3.1
Divide by .
Step 10
Find the period of .
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Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4
Replace with an approximation.
Step 10.5
Multiply by .
Step 10.6
Divide by .
Step 11
Add to every negative angle to get positive angles.
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Step 11.1
Add to to find the positive angle.
Step 11.2
Subtract from .
Step 11.3
List the new angles.
Step 12
The period of the function is so values will repeat every radians in both directions.
, for any integer