Trigonometry Examples

Find the Intersection of the Functions f(x)=tan(3x) , f(x)=0.5
,
Step 1
Substitute for .
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 2.3
Simplify the right side.
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Step 2.3.1
Evaluate .
Step 2.4
Divide each term in by and simplify.
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Step 2.4.1
Divide each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Cancel the common factor of .
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Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Divide by .
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Divide by .
Step 2.5
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 2.6
Solve for .
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Step 2.6.1
Add and .
Step 2.6.2
Divide each term in by and simplify.
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Step 2.6.2.1
Divide each term in by .
Step 2.6.2.2
Simplify the left side.
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Step 2.6.2.2.1
Cancel the common factor of .
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Step 2.6.2.2.1.1
Cancel the common factor.
Step 2.6.2.2.1.2
Divide by .
Step 2.6.2.3
Simplify the right side.
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Step 2.6.2.3.1
Divide by .
Step 2.7
Find the period of .
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Step 2.7.1
The period of the function can be calculated using .
Step 2.7.2
Replace with in the formula for period.
Step 2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 2.9
Consolidate and to .
, for any integer
, for any integer