Trigonometry Examples

Find the Intersection of the Functions f(x)=sin(4x) , f(x)=cos(4x)
,
Step 1
Substitute for .
Step 2
Solve for .
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Step 2.1
Divide each term in the equation by .
Step 2.2
Cancel the common factor of .
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Step 2.2.1
Cancel the common factor.
Step 2.2.2
Rewrite the expression.
Step 2.3
Convert from to .
Step 2.4
Rewrite the equation as .
Step 2.5
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 2.6
Simplify the right side.
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Step 2.6.1
The exact value of is .
Step 2.7
Divide each term in by and simplify.
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Step 2.7.1
Divide each term in by .
Step 2.7.2
Simplify the left side.
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Step 2.7.2.1
Cancel the common factor of .
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Step 2.7.2.1.1
Cancel the common factor.
Step 2.7.2.1.2
Divide by .
Step 2.7.3
Simplify the right side.
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Step 2.7.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.7.3.2
Multiply .
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Step 2.7.3.2.1
Multiply by .
Step 2.7.3.2.2
Multiply by .
Step 2.8
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 2.9
Solve for .
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Step 2.9.1
Simplify.
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Step 2.9.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.9.1.2
Combine and .
Step 2.9.1.3
Combine the numerators over the common denominator.
Step 2.9.1.4
Add and .
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Step 2.9.1.4.1
Reorder and .
Step 2.9.1.4.2
Add and .
Step 2.9.2
Divide each term in by and simplify.
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Step 2.9.2.1
Divide each term in by .
Step 2.9.2.2
Simplify the left side.
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Step 2.9.2.2.1
Cancel the common factor of .
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Step 2.9.2.2.1.1
Cancel the common factor.
Step 2.9.2.2.1.2
Divide by .
Step 2.9.2.3
Simplify the right side.
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Step 2.9.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.9.2.3.2
Multiply .
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Step 2.9.2.3.2.1
Multiply by .
Step 2.9.2.3.2.2
Multiply by .
Step 2.10
Find the period of .
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Step 2.10.1
The period of the function can be calculated using .
Step 2.10.2
Replace with in the formula for period.
Step 2.10.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.11
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer