Trigonometry Examples

Solve for x sin(x)+sin(2x)>0
Step 1
Apply the sine double-angle identity.
Step 2
Factor out of .
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Step 2.1
Raise to the power of .
Step 2.2
Factor out of .
Step 2.3
Factor out of .
Step 2.4
Factor out of .
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
The exact value of is .
Step 4.2.3
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 4.2.4
Subtract from .
Step 4.2.5
Find the period of .
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Step 4.2.5.1
The period of the function can be calculated using .
Step 4.2.5.2
Replace with in the formula for period.
Step 4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.5.4
Divide by .
Step 4.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Solve for .
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Divide each term in by and simplify.
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Step 5.2.2.1
Divide each term in by .
Step 5.2.2.2
Simplify the left side.
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Step 5.2.2.2.1
Cancel the common factor of .
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Step 5.2.2.2.1.1
Cancel the common factor.
Step 5.2.2.2.1.2
Divide by .
Step 5.2.2.3
Simplify the right side.
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Step 5.2.2.3.1
Move the negative in front of the fraction.
Step 5.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5.2.4
Simplify the right side.
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Step 5.2.4.1
The exact value of is .
Step 5.2.5
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 5.2.6
Simplify .
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Step 5.2.6.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.6.2
Combine fractions.
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Step 5.2.6.2.1
Combine and .
Step 5.2.6.2.2
Combine the numerators over the common denominator.
Step 5.2.6.3
Simplify the numerator.
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Step 5.2.6.3.1
Multiply by .
Step 5.2.6.3.2
Subtract from .
Step 5.2.7
Find the period of .
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Step 5.2.7.1
The period of the function can be calculated using .
Step 5.2.7.2
Replace with in the formula for period.
Step 5.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.7.4
Divide by .
Step 5.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate and to .
, for any integer
Step 8
Use each root to create test intervals.
Step 9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 9.1
Test a value on the interval to see if it makes the inequality true.
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Step 9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 9.1.2
Replace with in the original inequality.
Step 9.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 9.2
Test a value on the interval to see if it makes the inequality true.
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Step 9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 9.2.2
Replace with in the original inequality.
Step 9.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 9.3
Test a value on the interval to see if it makes the inequality true.
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Step 9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 9.3.2
Replace with in the original inequality.
Step 9.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 9.4
Test a value on the interval to see if it makes the inequality true.
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Step 9.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 9.4.2
Replace with in the original inequality.
Step 9.4.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 9.5
Test a value on the interval to see if it makes the inequality true.
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Step 9.5.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 9.5.2
Replace with in the original inequality.
Step 9.5.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 9.6
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
True
True
False
True
False
True
Step 10
The solution consists of all of the true intervals.
or or , for any integer
Step 11
Combine the intervals.
, for any integer
Step 12