Trigonometry Examples

Solve for x in Degrees 3tan(x)sin(x)=sin(x)
Step 1
Subtract from both sides of the equation.
Step 2
Simplify the left side of the equation.
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Step 2.1
Simplify each term.
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Step 2.1.1
Rewrite in terms of sines and cosines.
Step 2.1.2
Combine and .
Step 2.1.3
Multiply .
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Step 2.1.3.1
Combine and .
Step 2.1.3.2
Raise to the power of .
Step 2.1.3.3
Raise to the power of .
Step 2.1.3.4
Use the power rule to combine exponents.
Step 2.1.3.5
Add and .
Step 2.2
Simplify each term.
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Step 2.2.1
Factor out of .
Step 2.2.2
Separate fractions.
Step 2.2.3
Convert from to .
Step 2.2.4
Divide by .
Step 3
Factor out of .
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Step 3.1
Factor out of .
Step 3.2
Factor out of .
Step 3.3
Factor out of .
Step 4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
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
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
The exact value of is .
Step 5.2.3
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 5.2.4
Subtract from .
Step 5.2.5
Find the period of .
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Step 5.2.5.1
The period of the function can be calculated using .
Step 5.2.5.2
Replace with in the formula for period.
Step 5.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.5.4
Divide by .
Step 5.2.6
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
, for any integer
Step 6
Set equal to and solve for .
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Step 6.1
Set equal to .
Step 6.2
Solve for .
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Step 6.2.1
Add to both sides of the equation.
Step 6.2.2
Divide each term in by and simplify.
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Step 6.2.2.1
Divide each term in by .
Step 6.2.2.2
Simplify the left side.
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Step 6.2.2.2.1
Cancel the common factor of .
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Step 6.2.2.2.1.1
Cancel the common factor.
Step 6.2.2.2.1.2
Divide by .
Step 6.2.3
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 6.2.4
Simplify the right side.
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Step 6.2.4.1
Evaluate .
Step 6.2.5
The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 6.2.6
Add and .
Step 6.2.7
Find the period of .
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Step 6.2.7.1
The period of the function can be calculated using .
Step 6.2.7.2
Replace with in the formula for period.
Step 6.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.7.4
Divide by .
Step 6.2.8
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
, for any integer
Step 7
The final solution is all the values that make true.
, for any integer
Step 8
Consolidate the answers.
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Step 8.1
Consolidate and to .
, for any integer
Step 8.2
Consolidate and to .
, for any integer
, for any integer