Trigonometry Examples

Solve for x cos(pi/2+x)-sin(pi/2+x)=0
Step 1
Divide each term in the equation by .
Step 2
Cancel the common factor of .
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Step 2.1
Cancel the common factor.
Step 2.2
Rewrite the expression.
Step 3
Separate fractions.
Step 4
Convert from to .
Step 5
Divide by .
Step 6
Separate fractions.
Step 7
Convert from to .
Step 8
Divide by .
Step 9
Multiply by .
Step 10
Subtract from both sides of the equation.
Step 11
Divide each term in by and simplify.
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Step 11.1
Divide each term in by .
Step 11.2
Simplify the left side.
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Step 11.2.1
Dividing two negative values results in a positive value.
Step 11.2.2
Divide by .
Step 11.3
Simplify the right side.
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Step 11.3.1
Divide by .
Step 12
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 13
Simplify the right side.
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Step 13.1
The exact value of is .
Step 14
Move all terms not containing to the right side of the equation.
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Step 14.1
Subtract from both sides of the equation.
Step 14.2
To write as a fraction with a common denominator, multiply by .
Step 14.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 14.3.1
Multiply by .
Step 14.3.2
Multiply by .
Step 14.4
Combine the numerators over the common denominator.
Step 14.5
Simplify the numerator.
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Step 14.5.1
Multiply by .
Step 14.5.2
Subtract from .
Step 14.6
Move the negative in front of the fraction.
Step 15
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 16
Solve for .
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Step 16.1
Simplify .
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Step 16.1.1
To write as a fraction with a common denominator, multiply by .
Step 16.1.2
Combine fractions.
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Step 16.1.2.1
Combine and .
Step 16.1.2.2
Combine the numerators over the common denominator.
Step 16.1.3
Simplify the numerator.
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Step 16.1.3.1
Move to the left of .
Step 16.1.3.2
Add and .
Step 16.2
Move all terms not containing to the right side of the equation.
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Step 16.2.1
Subtract from both sides of the equation.
Step 16.2.2
To write as a fraction with a common denominator, multiply by .
Step 16.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.2.3.1
Multiply by .
Step 16.2.3.2
Multiply by .
Step 16.2.4
Combine the numerators over the common denominator.
Step 16.2.5
Simplify the numerator.
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Step 16.2.5.1
Multiply by .
Step 16.2.5.2
Subtract from .
Step 17
Find the period of .
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Step 17.1
The period of the function can be calculated using .
Step 17.2
Replace with in the formula for period.
Step 17.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 17.4
Divide by .
Step 18
Add to every negative angle to get positive angles.
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Step 18.1
Add to to find the positive angle.
Step 18.2
To write as a fraction with a common denominator, multiply by .
Step 18.3
Combine fractions.
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Step 18.3.1
Combine and .
Step 18.3.2
Combine the numerators over the common denominator.
Step 18.4
Simplify the numerator.
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Step 18.4.1
Move to the left of .
Step 18.4.2
Subtract from .
Step 18.5
List the new angles.
Step 19
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 20
Consolidate the answers.
, for any integer