Trigonometry Examples

Solve for x 5sin(3x)+6cos(3x)=1
Step 1
Use the identity to solve the equation. In this identity, represents the angle created by plotting point on a graph and therefore can be found using .
where and
Step 2
Set up the equation to find the value of .
Step 3
Evaluate .
Step 4
Solve to find the value of .
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Step 4.1
Raise to the power of .
Step 4.2
Raise to the power of .
Step 4.3
Add and .
Step 5
Substitute the known values into the equation.
Step 6
Divide each term in by and simplify.
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Step 6.1
Divide each term in by .
Step 6.2
Simplify the left side.
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Step 6.2.1
Cancel the common factor of .
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Step 6.2.1.1
Cancel the common factor.
Step 6.2.1.2
Divide by .
Step 6.3
Simplify the right side.
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Step 6.3.1
Multiply by .
Step 6.3.2
Combine and simplify the denominator.
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Step 6.3.2.1
Multiply by .
Step 6.3.2.2
Raise to the power of .
Step 6.3.2.3
Raise to the power of .
Step 6.3.2.4
Use the power rule to combine exponents.
Step 6.3.2.5
Add and .
Step 6.3.2.6
Rewrite as .
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Step 6.3.2.6.1
Use to rewrite as .
Step 6.3.2.6.2
Apply the power rule and multiply exponents, .
Step 6.3.2.6.3
Combine and .
Step 6.3.2.6.4
Cancel the common factor of .
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Step 6.3.2.6.4.1
Cancel the common factor.
Step 6.3.2.6.4.2
Rewrite the expression.
Step 6.3.2.6.5
Evaluate the exponent.
Step 7
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 8
Simplify the right side.
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Step 8.1
Evaluate .
Step 9
Move all terms not containing to the right side of the equation.
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Step 9.1
Subtract from both sides of the equation.
Step 9.2
Subtract from .
Step 10
Divide each term in by and simplify.
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Step 10.1
Divide each term in by .
Step 10.2
Simplify the left side.
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Step 10.2.1
Cancel the common factor of .
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Step 10.2.1.1
Cancel the common factor.
Step 10.2.1.2
Divide by .
Step 10.3
Simplify the right side.
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Step 10.3.1
Divide by .
Step 11
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 12
Solve for .
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Step 12.1
Subtract from .
Step 12.2
Move all terms not containing to the right side of the equation.
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Step 12.2.1
Subtract from both sides of the equation.
Step 12.2.2
Subtract from .
Step 12.3
Divide each term in by and simplify.
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Step 12.3.1
Divide each term in by .
Step 12.3.2
Simplify the left side.
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Step 12.3.2.1
Cancel the common factor of .
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Step 12.3.2.1.1
Cancel the common factor.
Step 12.3.2.1.2
Divide by .
Step 12.3.3
Simplify the right side.
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Step 12.3.3.1
Divide by .
Step 13
Find the period of .
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Step 13.1
The period of the function can be calculated using .
Step 13.2
Replace with in the formula for period.
Step 13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 14
Add to every negative angle to get positive angles.
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Step 14.1
Add to to find the positive angle.
Step 14.2
Subtract from .
Step 14.3
List the new angles.
Step 15
The period of the function is so values will repeat every radians in both directions.
, for any integer